UNIVERSITA DEGLI STUDI DI BARI “ALDO MORO”Dipartimento di Matematica

Dottorato di Ricerca in Matematica

XXVII Ciclo – A.A. 2014-2015

Settore Scientifico-Disciplinare:

MAT/07 – Fisica Matematica

Tesi di Dottorato

Spectral Methods in Random Matrix eory:from Classical Ensembles to antum Random Tensors

Candidato:Fabio Deelan Cunden

Supervisore della tesi:Prof. Paolo Facchi

Coordinatore del Doorato di Ricerca:Prof.ssa Addolorata Salvatore

Abstract

is thesis, Spectral Methods in Random Matrix eory: from Classical Ensembles to antumRandom Tensors, collects the research work done during these last years on various aspects ofRandom Matrix eory. e essay is tripartite.

Part I contains an introduction to the subject of Random Matrix eory, its fundamental ideas,its techniques and a few applications. Most of the material in this Part is in the random matrices“public domain” by now. Chapters 1 and 2 collect my personal understanding of the “state of theart“ and the methodological framework of the discipline. Chapter 3 contains some notions ofLarge Deviation eory. is Chapter is almost independent of the previous ones but it preparesthe ground for the subsequent Chapters.

e original material of the thesis is contained in Parts II and III. Part II is devoted to the“Coulomb Gas Method” and its applications. In Chapter 4 I present a shortened route (based onthermodynamical identities) for the evaluation of rate functions and free energies in 2D Coulombgas systems on the line, which eectively simplies the computations of the large deviation func-tions of random variables of the form tr a(Mn) where Mn belongs to a β-ensemble. Althoughthe mathematical content and the physical concepts are not new, an adequate translation inthe discipline of random matrices does not seem to have been formally published elsewhere. InChapter 5 I apply this improved Coulomb Gas Method to establish a joint large deviation principlefor spread measures (generalized and total variances of high-dimensional Gaussian datasets) inmultivariate statistics (eorem 5.1). A number of further results are derived, including nite sam-ple/variates formulas (Section 5.7) and new results in the classical seing of Statistics (Section 5.8).In Chapter 6 the very same technique is applied to the joint statistics transport observables(conductance and shot noise) in chaotic cavities supporting a large number of electronic channels.Currently available asymptotic results include the separate statistics of conductance or shot noisealone (computed explicitly by Vivo, Majumdar and Bohigas [222]-[223]). Using the improvedCoulomb gas technique I was able to deduce the joint asymptotic normality of conductance andshot noise and to produce the full phase diagram for this problem.

In Chapter 7 I present the derivation of a new universal formula for the covariance of linearstatistics An =

∑i a(λi) and Bn =

∑i b(λi) on β-ensembles of random matrices (eorem 7.14).

e formula relies on the universality of the smoothed two points kernel and on a “conformal mapmethod” for 2D Coulomb gases (Lemma 7.9). is covariance formula applies to one-cut matrixmodels and linear statistics with slow variation on the scale of the typical eigenvalues spacing.is formula generalizes earlier uctuation formulas for a single linear statistics by Dyson and

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Metha [70] and Beenakker [20]. Several new predictions (in form of Corollaries 7.1, 7.2, 7.3 and 7.4)are presented.

Part III is inspired from problems in the area of antum Information eory. is lastPart starts with Chapter 8 where I thoroughly review the main features of typical entanglementproperties of uniformly (i.e. unitarily invariant) distributed pure states in bipartite systems. isChapter contains two new results: the joint asymptotic distribution of the Renyi entropies foruniformly distributed pure states (eorem 8.8 and Corollary 8.9) and the joint law of the Schmidteigenvalues of a random state constrained to a submanifold of xed degeneracy of the Schmidteigenvalues (eorem 8.10).

e polarized ensembles introduced in Chapters 9 oer a way to break the rotational symmetry,eectively generically xing the purity (a standard entanglement quantier) of the reduced states,among other properties. e eect is obtained by superposing two quantum states sampledaccording to two distinct probability measures. is allows one to tune ensembles away fromuniform and favour rather less (or more) entangled state. e statistical properties of the purityare characterized explicitly for a one-parameter family of polarized states (eorems 9.1 and 9.3),and an algorithm is proposed for the generation of random pure states with xed local purity (seeExperiment 4). A number of formulas for the expectation of functions of the form tr f(ωA) arederived, where ωA = trB |ψ〉 〈ψ| is the reduced state of the random state |ψ〉 ∈HA ⊗HB undera polarized law. In some cases these results are powerful enough to allow for the proof of a limittheorem of the entire spectrum of ωA. ese original results (eorems 10.1, 10.5 and 10.6) arediscussed in details in Chapter 10.

is essay ends with Chapter 11 devoted to the extension of random matrices to mixturesof random tensors. Ambainis, Harrow and Hastings [6], inspired by questions formulated byLeung and Winter [141], have proved strong concentration bounds on sums of random tensorproducts of pure state projectors. I introduced a diagrammatic method to reproduce the momentcalculations of Ambainis et al., as well as a strengthening of their results to imply convergence ofthe spectrum of the random tensor with probability 1 to the Marcenko-Pastur law (eorem 11.11),thus proving the Conjecture 11.1 in [6]. I also developed a general framework for dealing withthis and other distribution laws in multi-partite systems, based on Lie group actions and freeprobability arguments. One of the main results of this Section is the proof (eorems 11.15, 11.16and 11.17) of the asymptotic freeness of random local unitary operators. Consequences of theseresults are the Generalized Marcenko-Pastur eorems 11.19, 11.20 and 11.21 (Generalized FreePoisson laws) .

A few, hopefully, tantalizing questions are le open in the Epilogue.I have tried to give due credit to any result of some importance. If there is no credit it is

implicit that I am the inventor.Some of the achievements presented here are already published (or pre-published). Chapter 5

is an extended version from [56]. e main result of Chapter 7 is published in [55]. Some ofthe results of Chapters 8 and 9 are reported in [53, 54]. e structural part of Chapter 4 and thecontent of Chapter 6 will appear in joint works with Paolo Facchi and Pierpaolo Vivo. e resultsof Chapter 8, 10 and 11 are not published yet. Other ndings [57, 58] are not reported in thisessay.

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Acknowledgments

First and foremost, thanks are due to Paolo Facchi. He has been a precious guide to me andmy debt to him is enormous. I was fortunate to meet and collaborate with him. I am obligedto Pierpaolo Vivo and Andreas Winter with whom I collaborated during my visiting periods atLaboratoire de Physique eorique et Modeles Statistiques - Universite Paris-Sud and Grup deInformacio antica at Universitat Autonoma de Barcelona. I am grateful to these two amazingpeople. I gratefully acknowledge both LPTMS and GIQ for hospitality and for providing aninspiring environment. e content of this thesis was also enriched by my cooperation withGiuseppe Florio. is thesis has improved with respect to structure and language through thecareful reading and commenting on dras of Paolo Facchi, Pierpaolo Vivo and Andreas Winter.Not least I want to thank my friends, colleagues and ocemates Ilaria Castellano, Sara Di Martinoand Giuseppe Stancarone with whom I shared this journey.

Contents

Preface vi

I RANDOM MATRIX THEORY 1

1 Random Matrix eory 21.1 Why random matrices are cool? . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 e classical ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 RMT: do it yourself. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Distribution of the eigenvalues at global scale . . . . . . . . . . . . . . . . 71.3.2 Behavior of the eigenvalues at local scales . . . . . . . . . . . . . . . . . 121.3.3 Concentration of measure for linear statistics . . . . . . . . . . . . . . . 15

2 Spectral Analysis of Large Dimensional Random Matrices 222.1 Moment method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Stieltjes transform method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Saddle-point method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.1 Coulomb gas analogy and Fekete points . . . . . . . . . . . . . . . . . . 282.3.2 Variational problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4 Free Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4.1 Basic denitions and results . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4.2 Free probability and Random Matrix eory . . . . . . . . . . . . . . . . 40

2.5 Levy’s metric and perturbation inequalities . . . . . . . . . . . . . . . . . . . . . . 412.6 Examples and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Notions of Large Deviations eory 483.1 Basic denitions and theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

II THE COULOMB GAS METHOD 53

4 e Coulomb gas method 54

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4.1 Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Large deviation functions for linear statistics . . . . . . . . . . . . . . . . . . . . 56

5 Large deviations of spread measures for Gaussian matrices 625.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2 Seing and summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.3 Scaled trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.4 Scaled log-determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.4.1 Smaller H : a dierent mechanism . . . . . . . . . . . . . . . . . . . . . . . 715.5 Joint cumulant GF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.6 Likelihood ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.7 Finite n,m results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.8 e classical seing in statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.9 Comparison between the log-gas and the noninteracting gas . . . . . . . . . . . 80

6 Joint statistics of quantum transport in chaotic cavities 836.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.2 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.3 e associated Coulomb gas problem . . . . . . . . . . . . . . . . . . . . . . . . 866.4 Solution of the variational problem . . . . . . . . . . . . . . . . . . . . . . . . . 896.5 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.6 Joint large deviation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7 Universal covariance formula 977.1 Seing and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987.2 Implications of the covariance formula . . . . . . . . . . . . . . . . . . . . . . . . 101

7.2.1 Asymptotic decorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017.2.2 Covariance formula in random matrix theory of quantum transport . . . 1027.2.3 Joint distribution of traces of the β-ensembles . . . . . . . . . . . . . . . 108

7.3 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.3.1 Some lemmas on conformal maps and Coulomb gas systems . . . . . . . . 1147.3.2 Solving kernel and universal covariance formula . . . . . . . . . . . . . . 117

III BEYOND THE CLASSICAL ENSEMBLES 122

8 e Unbiased Ensemble 1238.1 A short quantum-mechanical vocabulary . . . . . . . . . . . . . . . . . . . . . . 1238.2 Typicality for the unbiased ensemble . . . . . . . . . . . . . . . . . . . . . . . . . 1278.3 Random matrix theory of the unbiased ensemble . . . . . . . . . . . . . . . . . . 130

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9 Polarized Ensembles: typical purity 1379.1 Polarized ensembles of pure states . . . . . . . . . . . . . . . . . . . . . . . . . . 1389.2 One-parameter ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1399.3 Typical local purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

9.3.1 Local purity of one-parameter polarized ensembles . . . . . . . . . . . . 1409.3.2 Generation of random pure states with xed local purity . . . . . . . . . . 144

10 Polarized Ensembles: spectral properties 14710.1 Adding noise to a maximally entangled state . . . . . . . . . . . . . . . . . . . . 14810.2 Inject separability in the unbiased ensemble . . . . . . . . . . . . . . . . . . . . 15310.3 Superposition of maximally entangled states . . . . . . . . . . . . . . . . . . . . 158

11 Random Tensor eory 16111.1 Seing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16111.2 Summary of techniques and results . . . . . . . . . . . . . . . . . . . . . . . . . 16511.3 e Marcenko-Pastur law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

11.3.1 Gaussianization and Truncation tricks . . . . . . . . . . . . . . . . . . . 16611.3.2 e moment method for Wishart matrices . . . . . . . . . . . . . . . . . 16911.3.3 Some lemmas on graph theory and combinatorics . . . . . . . . . . . . . 170

11.4 e Ambainis-Harrow-Hastings ensemble and its generalization . . . . . . . . . . 17411.5 A Free Probability approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18011.6 Insights from Group eory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18011.7 A general problem and the Free Poisson law . . . . . . . . . . . . . . . . . . . . 18311.8 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18611.9 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

11.9.1 Sample covariance matrices . . . . . . . . . . . . . . . . . . . . . . . . . 18811.9.2 More generic positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Epilogue 193

References 199

List of Figures

1.1 Fundamental classication of self-adjoint ensembles. . . . . . . . . . . . . . . . 51.2 Output of Experiment 1: Wigner’s semicircle law. . . . . . . . . . . . . . . . . . 81.3 Output of Experiment 2: Wigner’s surmise. . . . . . . . . . . . . . . . . . . . . . . 141.4 Output of Experiment 3: concentration of measure. . . . . . . . . . . . . . . . . . 17

5.1 Scaer plot of data and their dispersions around the mean. . . . . . . . . . . . . 635.2 Rate function of the total variance. . . . . . . . . . . . . . . . . . . . . . . . . . . 695.3 Variance of H for c = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.4 Cumulant generating functions JT , JH and JL. . . . . . . . . . . . . . . . . . . . 77

6.1 2D Coulomb gas associated to the transmission eigenvalues. . . . . . . . . . . . 886.2 Phase diagram of condunctance and shot noise: Laplace and real space. . . . . . 92

7.1 e one-cut Coulomb gas systems as matrix models. . . . . . . . . . . . . . . . . 997.2 Decorrelation phenomenon for the traces of the Gaussian ensemble. . . . . . . . 1027.3 Covariance of the dimensionless conductance G and shot-noise P . . . . . . . . . . 1047.4 Contour of integration in the complex plane. . . . . . . . . . . . . . . . . . . . . 1057.5 Contour of integration in the complex plane around a cut. . . . . . . . . . . . . 1067.6 Covariance of the traces of the Wishart-Laguerre ensemble. . . . . . . . . . . . 109

8.1 Renyi’s entropies of the unbiased ensemble. . . . . . . . . . . . . . . . . . . . . 136

9.1 Typical purity of the polarized ensembles. . . . . . . . . . . . . . . . . . . . . . 146

10.1 Phase transition in the limiting support of a polarized ensemble. . . . . . . . . . 14910.2 Polarized ensemble: evaporation of the largest eigenvalue. . . . . . . . . . . . . . 15710.3 Generalized arcsine ensemble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

11.1 Bipartite graphs G(i,µ) and canonical representants ∆(`, r, s). . . . . . . . . . . 17111.2 Canonical graphs ∆(`, r, s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17211.3 Examples of (k + 1)-partite graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17711.4 Example of “rainbow diagram”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17811.5 Marcenko-Pastur law for random mixtures of tensors. . . . . . . . . . . . . . . . . 19111.6 Distribution of the sum of two rank-n/2 projections in generic position. . . . . 192

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List of Tables

7.1 Summary of various classical invariant matrix models. . . . . . . . . . . . . . . 100

11.1 Summary of the notation used for mixtures of random tensors I. . . . . . . . . . 16311.2 Summary of the notation used for mixtures of random tensors II. . . . . . . . . 183

v

Preface

Conventions and notation

For reasons of space, I will not be able to dene every single mathphysical term that I use in thisthesis. If a term is italicesed for reasons other than emphasis or for denition, then it denotes astandard object, result, or concept, which can be easily looked up in any pertinent textbook.

Set, structures and calculus If E is a set, then 1E is the indicator function of E: 1E(x) = 1 ifx ∈ E, and 0 otherwise. If F is a formula, then 1F is the indicator function of the set dened bythe formula F . e complement of a set E is denoted by Ec.

For any positive integer n, the set 1, 2, . . . , n will be abbreviated as [n]. We will concernmostly with the elds R, C and H (reals, complex and quaternion numbers respectively). Areference to the less familiar eld H is Appendix E in [8]. If z ∈ C, we denote by z∗ the complexconjugate of z.

e symbol log denotes the natural logarithm (the inverse of exp).

Function spaces C(X ) is the space of real valued continuous functions on X . By Cb(X ) wedenote the spaces of bounded continuous functions on X while Cc(X ) and C0(X ) are the spaceof continuous function compactly supported and vanishing at innity, respectively. All of themare equipped with the norm of uniform convergence ‖f‖∞ := sup|f |.

We use the standard notation Lp for the Lebesgue space of p-summable elements. e spaceand the measure will oen be implicit, but clear from the context.

Probability measures and random variables e symbol δx0 is the Dirac mass at x0.All probability measures considered are normalized Borel measure on Polish spaces, i.e. com-

plete, separable metric spaces, equipped with their Borel σ-algebras. A Borel measure µ isconcentrated on a set Λ if µ(Λc) = 0. By suppµ of a Borel (probability) measure µ we will denotethe essential support, i.e. the smallest closed set on which µ is concentrated. Note that suppµ isby denition a closed set. e set of probability measures µ with support contained in a domainΣ will be denoted byM(Σ). e same notation supp % will be used for the support of a function%. If Σ is a Polish space,M(Σ) is also a Polish space. e weak topology onM(Σ) is induced by

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convergence against Cb(Σ), i.e. bounded and continuous test functions. Hereaer,M(Σ) alwaysstands forM(Σ) equipped with the weak topology. e integral of a function f with respect to aprobability measure µ will be denoted indierently by

´dµ(x) f(x) or 〈µ, f〉.

e law of a random variable X will be denoted by law(X). We will stretch the language anddenote by law(X) any one among its probability measure, its distribution function or (if it exists)its probability density function. e mathematical expectation and variance of a random variableX will be wrien as E[X] and Var(X), respectively. By Cov(A,B) we denote the covarianceof two random variables A and B. A family of independent random variables (Xα)α∈A suchthat law(Xα) = law(Xβ) for any α, β is said i.i.d. (independent and identically distributed).A sequence of random variable Xn converges in distribution to the random variable X ifE[f(Xn)] converges to E[f(X)] for all f ∈ Cb(X ), i.e. if law(Xn) weakly converges to law(X).

e notation N (µ, σ2) denotes a Gaussian random variable with mean µ and variance σ2.e notation X ∼ N (µ, σ2) means law(X) = N (µ, σ2). More generally a sequence of randomvariables Xn is asymptotically normal with mean µn and variance σ2

n > 0 if (Xn − µn) /σnconverges in distribution to a N (0, 1) variable. We say that Xn is AN (µn, σ

2n). Here µn and

σ2n are sequences of numerical constants. Note that if Xn is AN (µn, σ

2n) it does not mean that

Xn converges in distribution to anything. However, for a range of probability calculations wemay treat Xn as a N (µn, σ

2n).

Asymptotic notation In most situations there will be an integer parameter n, which will oenbe going o to innity, and upon which most other quantities will depend; for instance we willoen be considering the spectral properties of n×n random matrices. Assuming that Y is positivewe write

X = O(Y )

X = o(Y )

X ∼ Yif X/Y

remains bounded

→ 0

→ 1 ,

as n→∞ (or in an asymptotic regime that should be clear from the context). We will oen useX ≈ Y to denote equivalence at logarithmic scales (logX ∼ log Y ).

e stochastic version of these notations have the following meaning. A random variable X(depending on a parameter n) is bounded in probability if for every ε > 0 there exists a constantKε such that Pr(|X| > Kε) ≤ ε in the asymptotic regime (for istance for suciently large n).e notation X = O(1) will be used. More generally X = O(Y ) denote X/Y = O(1), whileX = o(Y ) and X ∼ Y denote X/Y → 0 and X/Y → 1 (in probability), respectively.

Notation specic to random matrix theory For a generic n-uple λ = (λ1, . . . , λn) in Rn,the object µλ = 1

n

∑ni=1 δλi is the corresponding counting measure. If λi = λi(X) are the n

eigenvalues of an operator X , we shall write µX instead of µλ and call it the empirical spectraldensity of X (ESd of X). We will use very oen the Vandermonde determinant:

∆(λ) :=∏i<j

(λj − λi) = det(λj−1i

)i,j∈[n]

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Given a complex-valued function a, the map X 7→ An(X) = tr a(X) is referred as a linearstatistics dened by a. e corresponding scaled linear statistics is dened asA(X) = 1

nAn(X) =〈µX , a〉. Whenever not dangerous we use A[µX ] to denote A(X).

Notation specic to quantum mechanics All Hilbert spaces considered when dealing withquantum mechanical problems will be complex and nite-dimensional, with inner product 〈·|·〉,linear in the second entry. In this thesis we will oen use the Dirac notation [65]. If ψ is a nonzerovector of the Hilbert space H , then |ψ〉 := ψ/‖ψ‖ is a normalized vector in H , i.e. a vector onthe unit sphere S(H ). By duality, to every |ψ〉 corresponds one and only one linear functionalacting on H denoted by 〈ψ|. e linear projection operator along ψ is denoted by |ψ〉 〈ψ|. Forany linear operator A acting on H , its adjoint is denoted by A†. An operator such that A = A†

is called self-adjoint. We write A ≥ 0 and we say that the operator A is positive if 〈ψ |Aψ〉 ≥ 0for any ψ ∈ H . e set of linear (bounded) operators on H is denoted by B(H ). Positiveoperators with unit trace are called density operators. e set of density operators will be denotedby D(H ). In tensor product spaces HA ⊗HB , the partial trace trB over HB is the (unique)map from B(HA⊗HB) onto B(HA) satisfying trB (A⊗B) = tr (B)A for all A ∈ B(HA) andB ∈ B(HB). An analogous denition holds for trA . For more denitions and the language ofquantum information theory we refer to [228].

Numerics and simulations We will use histograms (or some variations) as estimators ofprobability density functions. In all of them the binwidths will be chosen in order to minimize themean integrated squared error (see Chapter 20 of [227]). Most of the simulations have been doneusing Mathematica and MATLAB.

Part I

RANDOM MATRIX THEORY

1

Chapter 1

Random Matrix eory

ere are many excellent books on Random Matrix eory (RMT). A beautiful pedagogical in-troduction to this subject is Topics in Random Matrix eory of Tao [211]. For a more detailedreview on the origin and the applications of RMT in nuclear physics and other related areas,we refer to the classical treatise Random Matrices by Mehta [156]. More developments of RMThave been recently collected in Handbook on Random Matrix eory by Akemann, Baik and DiFrancesco [2]. e literature on the methodologies and mathematical aspects of RMT is verylarge. Among this huge supply of books and tutorials, some of them deserve a special mention:An introduction to random matrices by Anderson, Guionnet and Zeitouni [8], Spectral Analysisof Large Dimensional Random Matrices of Bai and Silverstein [12], Orthogonal Polynomials andRandom Matrices: A Riemann-Hilbert Approach by Dei [61] and the already mentioned bookof Mehta [156]. Although the initial progress achieved in RMT with the initial methods (dia-grammatic, eld-theoretical. . . ) has been impressive, a lot of very dierent techniques have beengradually introduced and developed. is RMT technology comprises for instance Fredholmdeterminants and determinantal processes [120, 203, 85], integrable systems [13] and Riemann-Hilbert problems [61]. At the moment, free probability provides the more abstract approach toRMT. e concept of “freeness” introduced by Voiculescu [226, 225] around 1986 in the theory ofoperator algebras, has revealed unexpected connections and implications with large dimensionalrandom matrices. Hopefully this new discipline could provide a conceptual understanding of thesurprising eects in spectral theory of random operators. In [211, 2, 8, 12] there are some RMToriented notions of free probability. A more specialized material are the reviews of Biane [26]and Nica and Speicher [166, 204].

It is evident from this brief (and incomplete!) recollection that Random Matrix eory is not asharply-dened discipline, with countless applications that deserve themselves special aention.

Although the material of this chapter is introductory and elementary, there are some explicitexamples which are not simple to trace in the existing literature and do not seem to have beenformally published elsewhere. e presentation is quite informal in the sense that I will not tryto present the various statements in full rigor or generality. I will rather aempt to outline themain concepts, ideas and techniques preferring a good illuminating example to a general proof. A

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3

more rigorous and detailed exposition is postponed to the next chapters.

1.1 Why random matrices are cool?

What is a random matrix? In order to dene this concept, we need a probability law on some setof matrices. If one has in mind a nite set of matrices, there is an obvious way to draw an elementat random: choose each matrix with equal probability. More generally, if the set of matrices isa compact group (e.g., the group of n by n complex unitary matrices) there is again a naturalnotion of a random matrix obtained by using the normalized Haar measure on the group as theprobability measure. If the set is not compact (e.g., n by n self-adjoint matrices), then we haveto specify a suitable probability measure. One aspect of “universality” that is referred to in therandom matrix literature is that the asymptotic behavior of, for example, the eigenvalues is robust,in the sense that a number of reasonable probability models give the same behavior.

e study of random matrices originated in statistics, with the investigation of sample co-variance matrices [237]. However, the commonly cited origins of the “random matrix” ideas arein Wigner’s work on the scaering of neutrons o heavy nuclei [232, 233, 234, 235, 236]. eobserved energy levels in the scaering of many-body quantum system are the eigenvalues of aSchrodinger operator. Wigner had the marvelous idea that the large eigenvalues of a Schrodingeroperator should be distributed in a way very close to the distribution of the eigenvalues of arandom large Hermitian matrix. e accuracy of this stochastic model of the eigenvalues isstriking, and the success of this idea has led to its proliferation in many parts of physics dur-ing the last y years. For instance, in Wigner’s context the random behavior arises from thecomplicated many-body quantum system, but it is now known that this behavior is exhibited bychaotic quantum systems even if there are only a few degrees of freedom. More recently, therandom matrix philosophy has come up in various areas of mathematics, and one has the distinctfeeling that this is an emerging paradigm of great importance. is natural blend of classicalprobability and spectral theory is denitely one of the most protable manifestations of a general“noncommutative way of thinking” [211] that permeates modern mathematics.

1.2 e classical ensembles

We turn to the initial question, what is a random matrix? Maybe the best denition is the simplestone:

Denition 1.1. A random matrix is a matrix whose entries are random.

e matrices usually considered in RMT have real, complex or quaternionic entries. esechoices correspond to Dyson’s threefold way [69]. We shall consider in particular real symmetric,complex Hermitian and quaternion self-dual matrices. In the following we will denote them withthe generic term “self-adjoint random matrices”. We do not, however, want to look at completelyarbitrary random matrices. e following classication singles out two particularly aractive anduseful categories of random matrix ensembles:

4

1. Wigner matrices: random matrices whose entries are independently distributed (modulothe self-adjointness condition). e law of a Wigner matrix Mn = (Mij)i,j∈[n] is speciedby the individual distributions of the entries of the upper-triangular part:

law(Mn) =∏

1≤i≤j≤nlaw(Mij) . (1.1)

2. Invariant ensemble: these ensembles are characterized by invariance under “rotations”,i.e.

law(Mn) = law(UMnU−1) . (1.2)

e matrices U belong to the orthogonal group (for real symmetric Mn), to the unitarygroup (for complex Hermitian Mn) or to the symplectic group (for quaternion self-dualMn). Here and in the sequel we will use the word “unitary” independently of the underlyingeld. A lemma due to Weyl [229] states that the rotational invariance of the measure (1.2)can be realized if and only if the law of the n-dimensional matrix Mn is specied by thejoint law of the traces of its rst n powers:

law(Mn) = law(trMn, trM2n, . . . , trM

nn ) . (1.3)

A less general invariant law (but oen manageable) has the form

law(Mn) = const× e−trV (Mn)dMn , (1.4)

where dMn is the at Lebesgue measure on the entries of Mn, and V is a real-valuedfunction.

Having in mind the probability theory of scalar random variables one could think that ensembleswith independent entries are simpler to analyze. Surprisingly, usually this is not the case! However,the milestone results in RMT have been established for a very special ensemble with independententries, namely self-adjoint matrices with independent Gaussian entries (with the same mean andvariance). In RMT the Gaussian paradigm, not only provides the standard advantages of Gaussianrandom variables but has an additional crucial bonus: the Gaussian ensemble is the intersectionof the two main categories of self-adjoint random matrices.

eorem 1.1 (Porter-Rosenzweig [185, 186]). Let Mn = M †n be a n-dimensional random matrixwhose law satises simultaneously the requirements:

(i) law(Mn) =∏

1≤i≤j≤nlaw(Mij);

(ii) law(Mn) = law(UMnU†), for every unitary U .

en

law(Mn) = const× e−[a trMnM

†n+b trMn

]dMn , (1.5)

for some constants a > 0 and b ∈ R.

5

i.i.d. entries

Unitary invariant

Gaussian

P(Mn) =∏ij p(Mij)

P(Mn) = P(UMnU−1)

P(Mn) ∝ e−trV (Mn)

Figure 1.1: Fundamental classication of self-adjoint ensembles.

For every matrix Mn = M †n, we have at hand the natural parametrization (Mij)i≤j . However,for a unitarily invariant random matrix, it is not simple to write down the probability measureof Mn in terms of these coordinates. However, a standard result of linear algebra provides aconvenient set of coordinates. Every n×n self-adjoint matrixMn admits a spectral decompositionMn = UDU †, with U the unitary matrix of eigenvectors and D = diag(λ1, . . . , λn) the realeigenvalues of Mn. A standard computation shows that the real codimension of the set of self-adjoint matrices with exactly two equal eigenvalues is equal to β + 1, with β = 1, 2, or 4 for real,complex or quaternion matrices respectively.

Example 1.1 (Higher codimension: Hermitian case). Let Mn be a n × n Hermitian matrix(complex entries, β = 2). In the spectral representation: Mn = UΛU †, where U ∈ U(n) is an-dimensional unitary matrix, and D = diag (λ1, . . . , λn) is the diagonal matrix of eigenvaluesof Mn. It is not dicult to prove that, typically, the spectrum of Mn is simple, i.e. the set ofHermitian matrices with simple spectra is open and dense has a manifold in Rn2 . Another naturalquestion is: “given Mn and D, how many (real) degrees of freedom does U have?”. Although theanswer is intuitive, from a detailed derivation we can obtain a precious result. Let us suppose:

∃U1, U2 s.t. Mn = U1DU†1 = U2DU

†2 .

en(U †2U1)D = D(U †2U1) =⇒ [(U †2U1), D] = 0 ,

6

V = U †2U1 = (vjk)j,k∈[n] commutes with the eigenvalues diagonal matrixD = diag (λ1, . . . , λn):v11 v12 . . . v1n

v21 v22 . . . v2n...

... . . . ...vn1 vn2 . . . vnn

λ1 0 . . . 00 λ2 . . . 0...

... . . . ...0 0 . . . λn

=

λ1 0 . . . 00 λ2 . . . 0...

... . . . ...0 0 . . . λn

v11 v12 . . . v1n

v21 v22 . . . v2n...

... . . . ...vn1 vn2 . . . vnn

,

(1.6)i.e. vjkλk = λjvjk . So, ifMn has a simple spectrum (distinct eigenvalues), vjk = 0 for j 6= k: V isa diagonal unitary matrix. en, a matrixU that diagonalizesMn belongs to U(n)/Tn, a (n2−n)-dimensional real set (we denote by Tn the n-dimensional torus with T = z ∈ C : |z| = 1 ).

What happens if Mn has at least one degenerate eigenvalue, say λ1 = λ2? en, V has theblock form ( )v11 v12 0

v21 v22

0 Diag, (1.7)

and belongs to a subgroup of U(n). In fact, the matrices U ’s that diagonalizes Mn belong toU(n)/(set of matrices of type V ), of dimension n2 − (n + 2). e diagonal matrix D has now(n−1) independent real parameters. erefore the submanifold ofMn’s with at least one repeatedeigenvalue has dimension n2 − (n− 2) + (n− 1) = n2 − 3, i.e. codimension 3 (equal to β + 1).

Observe that, naively one would expect a codimension 1 for the set of self-adjoint matriceswith one repeated eigenvalue. In fact, this higher codimension of non-simple matrices suggests arepulsion phenomenon: since it is very unlikely for the eigenvalues to be equal, there must besome force that repels the eigenvalues of self-adjoint (and in general normal) matrices hinderingthem from geing too close to each other. is intuition is conrmed by the following theoremthat claries why invariant ensembles are somehow simpler from a spectral viewpoint.

eorem 1.2. Let Mn be a self-adjoint n× n random matrix with unitarily invariant law

law(Mn) = const× e−trV (Mn)dMn , (1.8)

where β = 1, 2 or 4 ifMn is real symmetric, complex Hermitian or quaternion self-dual respectively,and V a real-valued function. If we write Mn in terms of ts spectral decomposition Mn = UDU †

we have that:

(i) U and D are independent;

(ii) U is distributed according to the Haar measure on the unitary group;

(iii) the joint probability density function of the eigenvalues λ = (λ1, . . . , λn) is

P(λ) = const× e−∑k V (λk)

∏i<j

|λi − λj |β . (1.9)

7

e beauty and the consequences of this theorem in RMT are striking and deserve a fewcomments. Firstly, eigenvectors and eigenvalues of a unitarily invariant random matrix X areindependent. Moreover, informally speaking, the eigenvectors of a unitarily invariant matrixare quite uninteresting because they can be with equal probability any of the orthonormal basesof the n-dimensional space. is is not surprising, since the unitarily invariance is exactly therequirement that there is no preferred basis for the system. What is really surprising, is that weget access to the precise law of the eigenvalues ofX . e passage from the canonicalMn = (Mij)to the polar (λk, Uij) coordinates has the price of the Jacobian factor

|∆(λ)|β =∏i<j

|λi − λj |β , (1.10)

that accounts for the eigenvalue repulsion. is repulsion (that has a purely geometric nature) isindeed responsible of many intriguing phenomena in RMT.

For Wigner matrices we know the exact distribution of the entries, but a change of coordinatesimmediately makes this knowledge immaterial. A change of basis modies the distribution ofthe entries in a way that, in general, we are not able to control. In particular, we do not knowhow to get the eigenbasis and what is the distribution of the entries in this special basis (i.e. thejoint law of the eigenvalues). On the contrary, and in some sense counter-intuitively, with thecoordinate-free description of the unitarily invariant ensemble (1.4) we have the distribution ofthe entries in any bases and in particular we know exactly the law of the eigenvalues.

During a talk in the 90s, Alan Edelman showed a few examples to the audience to answerthe question “Why eigenvalues of random matrices are cool?”. Aer many years, this subject isstill fascinating and has been developed a lot. Moreover, a lot of aention is now deserved byeigenvectors of random matrices in connection to localization phenomena for random Schrodingeroperators [79]. So, why spectral analysis of random matrices is interesting? I will present a fewphenomena that arise when dealing with random matrices that should convince everyone aboutthe coolness of RMT.

1.3 RMT: do it yourself.

Although the advantage of using RMT lies in the possibility of computing explicit mathematicaland physical quantities analytically, it is sometimes necessary to resort to numerical simulations.Since the best way to understand a phenomenon is by experiments, we will provide three (veryeasy) ones 1. Aer each experiment, some related concepts and motivations are introduced.

1.3.1 Distribution of the eigenvalues at global scale

Hereaer we will concern with the behavior of the (random) eigenvalues of a random matrix. Oneof the milestone of RMT is the robustness of the empirical spectral distribution of the eigenvalue.What does it mean? In the following experiment, we numerically diagonalize an n × n real

1Excellent references on numerical experiments on random matrices are [73, 74]

8

symmetric random matrix Mn with Gaussian entries. One should perform the experiment forincreasing values of n and then analyze the dierent outputs. ere is no hope to know the exactlocation of the eigenvalues λ = (λ1, . . . , λn) of a random matrix for large n. However, if n isvery large, the eigenvalues tend to be concentrated in a bounded region and to be distributedwith a well-dened distribution. is is the easiest way to contemplate a ne occurence ofconvergence of the spectral distribution of a random matrix model. In Fig. 1.2 it is shown theoutput Histogram[λ] for dierent values of n.

Experiment 1 Global distribution of the eigenvalues and Wigner’s law1: (∗ Construct an n× n matrix and ll it with i.i.d.random numbers ∗)2: for i = 1, . . . , n do3: for j = 1, . . . , n do4: Aij = Gauss(0, 1)5: end for;6: end for;7: Mn =

(A+A†

)/√

2n; (∗ Take the scaled symmetric part of A ∗)8: λ = Eigenvalues[Mn]; (∗ Diagonalize W ∗)9: Histogram[λ]. (∗e output is the histogram of the eigenvalues of Mn ∗)

-2 0 2Λ

n = 10

-2 0 2Λ

n = 100

-2 0 2Λ

n = 1000

Figure 1.2: Output Histogram[λ] (normalized histogram of the eigenvalues of Mn) of Experiment 1 for dier-ent values of the matrix model n = 10, 100, 1000. As n increases Histogram[λ] approaches the Wigner’s law(2π)−1

√4− λ21|x|<2. is is a practical exercise to see the almost sure convergence of the spectral density of the

Gaussian ensemble.

It is now a good time to introduce some key objects in the investigation of the spectralproperties of random matrices.

Denition 1.2. Let Mn be a (possibly random) self-adjoint operator acting on a n-dimensionalspace. We dene the following objects:

µMn =1

n

∑i∈[n]

δλi(Mn) , (1.11)

FMn(t) =# λi(Mn) ≤ t

n=

ˆ t

−∞dµMn(x) , (1.12)

9

where λi(Mn) are the (necessarily real) eigenvalues of Mn. From now on we will call µMn theempirical spectral density (or ESd for short) and FMn the empirical spectral distribution (or ESD)of the operator Mn.

We will use the following notational convention: for a measure µ on R and a function f(x),we write

〈µ , f〉 :=

ˆR

dµ(x) f(x) . (1.13)

When no confusion is possible we will denote the eigenvalues of Mn by λi for simplicity andsometimes we will use the shortened notations µn ≡ µMn , Fn(t) ≡ FMn(t). e ESd is thediscrete measure on the line µn ∈ M(R) that puts a mass 1/n on each eigenvalue of W . efunction Fn(t) is the corresponding cumulative distribution function. We need to introduce afurther concept:

Denition 1.3. Let Mn denote a random self-adjoint operators acting on a n-dimensional spaceand let a be a complex-valued function dened on the spectrum of Mn. e complex-valued mapAn = tr a(·) is called linear statistics on the random matrix model Mn dened by a. Concretely:

An(Mn) = tr a(Mn) =∑i∈[n]

a(λi) . (1.14)

e empirical spectral density is a central object in the study of random matrices. For example,one easily veries the following identities:

FW (t) =

ˆdµMn(x) θ(t− x) , ∀t ∈ R , (1.15)

1

ntrWm =

ˆdµMn(x)xm , ∀m ∈ Z , (1.16)

and, in general, any linear statistics An(Mn) = tr a(Mn) can be wrien as an expectation withrespect to the probability measure µMn

An(Mn) = n 〈µMn , a〉 (1.17)

(equation (1.16) corresponds to the choice a(x) = xm). Oen we will rescale the linear statisticsas A(Mn) := 1

nAn(Mn). We have

A[µMn ] ≡ A(Mn) = 〈µMn , a〉 . (1.18)

e basic observation (1.16) that the moments of the ESd µn are the normalized traces of powersof Mn is the starting point of the so-called method of moments (see the next chapter).

WhenMn is a random matrix (as the one of Experiment 1), the ESd µn is a random probabilitymeasure - i.e. a random variable taking values in the spaceM(R) of probability measures onthe real line (equivalently, the ESD Fn is a random function depending in a complicated mannerby the law of the entries of Mn). Of particular interest is the case of very large n, since it oen

10

appears that for n→∞ the random quantity Fn(t) converges to a nonrandom number. Considerthe behaviour of the ESd of a sequence of Hermitian matrices Mn as n → ∞. Recall that forany sequence of random variables in a σ-compact metrizable space, one can dene notions ofconvergence in probability and convergence almost surely. Specializing these denitions to thecase of random probability measures on R, and to deterministic limits, we say that a sequenceof random ESd µn converges in probability (resp. converges almost surely) to a deterministic lawµ ∈ M(R) if, for every test function ϕ ∈ Cb(R) (continuous and bounded), the numericalquantities 〈µn , ϕ〉 converges in probability (resp. almost surely) to 〈µ , ϕ〉.

As usual, convergence almost surely implies convergence in probability, but not vice versa.ere is an even weaker notion of convergence, namely convergence in expectation, dened asfollows. Given a random ESd µn on can dene its expectation Eµn ∈M(R) via duality as

〈Eµn , ϕ〉 := E 〈µn , ϕ〉 , ∀ϕ ∈ Cb(R) . (1.19)

We then say that µn converges in expectation to a limit µ ∈M(R) if Eµn(λ) weakly convergesto µ.Remark 1.1. If the measures are tested against continuous and compactly supported functionsϕ ∈ Cc(R) we can predicate at most the convergence in the vague topology. A classical examplethat show the nonequivalence of the two notions is the sequence of singular measures µn = δn(n ∈ N). It is immediate to see that δn converges vaguely to the null measure (which is not aprobability measure!). However δn does not converge weakly. In the spectral analysis of randommatrices it is sometimes sucient to focus on the vague converges (compactly supported testfunctions) and then some confusion can arise in the literature. In fact, many random matrixmodels of interest have limit spectral density with compact support and the avenue of “escape ofinnity” (as in the example) does not occur.

We summarize the above notions of convergence.

Denition 1.4 (Modes of convergence). Consider a sequence Mn of random hermitian matricesand the corresponding sequence of empirical spectral densities µn. Let µ be a probabilitymeasure inM(R).

(i) µn converges almost surely to (or converges with full probability to) µ if for all ϕ ∈ Cb(R)

Pr (〈µn , ϕ〉 → 〈µ , ϕ〉) = 1 ; (1.20)

(ii) µn converges in probability to µ if for all ϕ ∈ Cb(R) and for every ε > 0

limn→∞

Pr (|〈µn , ϕ〉 − 〈µ , ϕ〉| > ε) = 0 ; (1.21)

(iii) µn converges in expectation to µ if

〈Eµn , ϕ〉 → 〈µ , ϕ〉 , ∀ϕ ∈ Cb(R) . (1.22)

11

In general, these notions of convergence are distinct from each other; but in practice, oneoen nds for many ensembles of random matrices that these notions are eectively equivalentto each other thanks to the concentration of measure phenomena. One says that an empiricalspectral density is a self-averaging object in the large dimensional limit. For this reason, in the lastyears many methods have been developed in order to compute the macroscopic (large n) averagespectral density for a variety of ensembles. Whenever it exists, we will call the nonrandom limitµ = limnE[µMn ] the density of states of the ensemble. What is the relation between the densityof states and the joint law of the eigenvalues of a random matrix? For the invariant ensemblesthe answer is immediate. Indeed, evauating the average over P(λ1, . . . , λn):

〈Eµn , ϕ〉 = E 〈µn , ϕ〉

=

ˆ n∏j=1

dλjP(λ1, . . . , λn)

ˆdλ

1

n

∑i∈[n]

δλi(λ)ϕ(λ)

=1

n

∑i∈[n]

ˆ ∏j∈[n]j 6=i

dλjP(λ1, . . . , λi−1, λi, λi+1, . . . , λn)ϕ(λi)

=

ˆdλP(1)(λ)ϕ(λ) . (1.23)

In the last line we have used the fact that the eigenvalues λi’s are exchangeable. is prove thatthe average spectral density is nothing but the one-body marginal distribution of the joint law ofthe eigenvalues.

Coming back to our Experiment, we observed that, as the size of the random matrix Mn

increases, Histogram[λ] (which is an estimator of µn) approaches a limiting shape. In thelanguage introduced so far, when the size of the random matrix increases, the ESd ofMn convergesweakly almost surely to a nonrandom distribution. is nonrandom distribution is the celebratedWigner’s semicircle law dµsc(x) = %sc(x)dx with

%sc(x) =1

2π

√4− x21|x|<2 (1.24)

(the shape in fact is a semiellipse). Understanding “why” is one of the jobs of RMT. In orderto stay closer to the experimental avour of this Chapter we oer an experimental meaning ofthe dierent modes of convergence. About the convergence in expectation, we mean that forany bounded and continuous test function ϕ if we take many realizations of Mn, the average oftrϕ(Mn) (over the randomness of Mn) converges to 〈µsc , ϕ〉. Experimentally this occurrencecorresponds to the following phenomenon: the average Histogram[λ] over many realizations ofMn (at xed n), becomes more and more indistinguishable from %sc(x) as we increase the size nof the random matrices (and the number of realizations of Mn). In fact Experiment 1 shows astronger form of convergence: for any bounded continuous function ϕ, for almost any sequencetrϕ(Mn), these values converge to 〈µsc , ϕ〉. is is the meaning of almost sure convergence.In our case, as we increase the size of the random matrices Mn, the (single realization) histogramof the eigenvalues Histogram[λ] will look like the Wigner’s law almost surely.

12

Example 1.2 (A random matrix model whose ESd converges in expectation but not almost surely).It is always instructive to have counterexamples in mind. Here is a simple random matrix whoseESd converges in expectation to a deterministic limit but does not have a deterministic limit in thealmost sure sense. Consider the n dimensional random matrixXn dened as Pr(Xn = In) = 1/2and Pr(Xn = −In) = 1/2, where In is the n × n identity matrix. One can easily show thatEµXn = (δ−1 + δ1) /2 (for any n) and then µXn converges in expectation to a measure equallyconcentrated in x = −1 and x = 1. However, µn does not approach any deterministic limit inthe almost sure sense, not even in probability.

e ensemble of random matrices considered in Experiment 1 is referred in literature as aWigner ensemble, i.e. a random matrix model with i.i.d. entries, modulo the symmetry constraint.In fact, so far we considered a specic Wigner ensemble by choosing real normally distributedentries. is ensemble is classically known as the GOE ensemble (Gaussian Orthogonal Ensemble).Other related ensembles are the GUE (Gaussian Unitary Ensemble) and the GSE (GaussianSymplectic Ensemble) whose construction is equal to that of the GOE except from the factthat the entries of the random matrix are complex or quaternions (more in the next chapters).

What happens if we change the distribution of the entries instead of using independentGaussian variables? Is the Wigner semicircle law peculiar of the GOE (and the other Gaussianensemble)? A stronger version of Wigner’s theorem [8] states that a large class of Wigner (i.i.d.entries) random matricesMn with the same rst and second moment of the Gaussian model admitsthe Wigner’s semicircle law %sc(λ) as limiting spectral density. is is one of the centerpieceuniversality results in random matrix theory and one of the reasons why (a posteriori) all randommatrix techniques and ideas can be most clearly and consistently introduced using the Gaussiancase as a paradigmatic example.

1.3.2 Behavior of the eigenvalues at local scales

e Wigner’s semicircle law (and its analogues for dierent random matrix ensembles) hastraditionally been among the rst and most elementary results established on random matrices.e empirical density is shown to converge weakly on macroscopic scales, i.e., on intervals thatcontain O(n) eigenvalues. However the semicircle law holds on much smaller scales as well (forthese estimates on local scales see for instance [105, 78]). We will not discuss this fascinatinglocal semicircle law, which is a very active topic nowadays. Instead, in the interpretation ofHermitian random matrices as random Hamiltonians we can concern about the distribution ofthe energy levels, i.e. the nearest neighbor spacing distribution pn(s). What is the probabilitythat two consecutive eigenvalues have spacing s? e question is undoubtedly well-posed, butthe answer is highly non-trivial! Moreover, if one suspects the emergence of a new type ofuniversality phenomenon, and want to verify it by an experiment, a naive implementation ofthis programme will not work. Indeed, the “universality” of the spectral correlations should beindependent of the spectral density (which is, in general, peculiar of the ensemble). en, beforelooking at the spacing between the eigenvalues of a random matrix, it is necessary to performthe so-called unfolding procedure. Roughly speaking, knowing the limiting spectral density of

13

a matrix ensemble, one is able to “unfold” the spectrum λ1, . . . , λn →λ

(u)1 , . . . , λ

(u)n

and create a new sequence of levels uniformly distributed. ese unfold levels exhibit a quite“universal” nearest-neighbor spacing distribution which is well approximated by the so-calledWigner’s surmise (see the output of Experiment 2):

pW (s) ' π

2se−

π4s2 , (1.25)

guessed by Wigner from a 2× 2 computation and tested against experimental data many times.

Experiment 2 Level spacing distribution and Wigner’s surmise

1: Dene the function Fsc(t) =´ t−∞ dx %sc(x) with %sc(x) = (2π)−1

√4− x2.

2: Construct an n × n random matrix Mn as in Experiment 1 (GOE); for simplicity take nmultiple of 4;

3: λ = Eigenvalues[Mn]; (∗ if not automatic, arrange the eigenvalues in increasing order ∗)4: λ(u) = Unfold[λ]; (∗ Unfolding by: λ(u)

i = nFsc(λi) ∗)5: for k = 1, . . . , n/2 do6: sk = (λ

(u)n4

+k − λ(u)n4

+k−1) , (∗ Level spacing in the bulk∗)7: end for8: Histogram[s].

Is there universality of these shapes with respect to small changes in the distribution of the matrixentries? Surprisingly (or not), the answer is ‘yes’.

e nearest-neighbor spacing distribution inherits its universality inRMT from the universalityof the pair correlation function (see the next subsection). e Wigner’s surmise has been oneof the rst manifestations that random matrices could model the behavior of a system by anappropriate set of matrices. We oer here a representative sample of results and applications ofRMT.

Energy levels in nuclear physics e rst physical motivation to investigate random matricesis as a model of the Hamiltonian H of a disordered quantum system. Here the threefold wayof Dyson has a precise physical meaning. e symmetries of H stem from this consideration:real symmetric matrices represent Hamiltonians of system with time reversal invariance (i.e. nomagnentic eld); complex Hermitian matrices correspond to systems without this symmetry;quaternion self-dual matrices describe systems with odd-spin and no rotation symmetry. Foran exhaustive account see [156]. Wigner originally invented random matrices to mimic theeigenvalues of the unknown Hamiltonian of heavy nuclei; lacking any information, he assumedthat the matrix elements (he was considering only bound states) are random variables subjectto the self-adjointness condition. He argued that, although this crude approximation couldnot predict the individual levels of the nucleus, some features of the statistics of the energylevels are shared by many nuclei and should be captured within this minimal model. Indeed,comparing experimental data with numerical computations of certain random matrices, he found

14

0 1 2 3 4s

pHsL

n = 7000

Figure 1.3: Histogram of the of level spacing in the bulk of a n× n matrix of the GOE ensemble (see Experiment 2).e dashed blue line is the Wigner’s surmise pW (s) (1.25).

that the energy gap statistics showed remarkable coincidence and robustness. In particular, heobserved that the energy levels of nuclei tend to “repel” each other. is repulsion, peculiar of theeigenvalues of random matrices, produces a signicant dierence from the gap statistics of fullyuncorrelated random points (Poisson process), and is in fact compatible with what nowadays wecall Wigner’s surmise.

antum chaos In classical mechanics, if a system is integrable the trajectories are connedon an invariant torus dened by the conserved quantities. However, we know that even somesimple systems can exhibit a chaotic dynamics: the phase-space cannot be parametrized withangle-action variables (because there are not enough conserved quantities), and the system visitsergodically a large portion of the phase space. A paradigmatic example of classical system thatcan exhibit both behaviors (integrability and chaos) is a classical billiard. Depending on the shape,some billiards are completely integrable (i.e. the ellipse, the rectangle), others are chaotic (i.e.the stadium, the Sinai billiard). Recently, it has been possible to miniaturize such billiards atmesoscopic scales (order of 1µm). e billiard is a small, at and smooth cavity in a semiconductormaterial, and the billiard ball is an electron that collides elastically with the boundary of the cavity.e cavity is kept at very low temperature, so that the wavelength of the electron is much largerthen the size of the cavity, and the motion can be considered ballistic. e question is: can we see

15

a “chaotic” behavior for quantum systems? e answer seems to be positive. Indeed, if the billiardis classically chaotic, the energy levels of the quantum systems (the miniaturized billiard) repeleach other as we vary a parameter of the system (like a magnetic eld, or the temperature). Onthe contrary, if the billiard is integrable, the energy levels ignore each other and there are manycrossing. is level repulsion seems to be a signature of the candidate quantum chaotic systems.Actually, this repulsion is also a characteristic feature of the eigenvalues in RMT. is is why RMTfound a lot of interest for quantum chaos. In this case, though the Hamiltonian is not random (itis given), it was observed and then conjectured by Bohigas, Giannoni and Schmit [28] that whenthe corresponding classical problem is chaotic, the energy levels are distributed like the spectrumof a random matrix. is is why in quantum chaos oen people consider random Hamiltonians(usually from the Gaussian ensemble corresponding to the symmetries of the system).

Interestingly, the next development in the area has (apparently) nothing to do with physics.

Riemann’s conjecture It is conjectured that the nontrivial zeros of the Riemann’s zeta function

ζ(s) =∑n≥1

n−s , (1.26)

all lie on the line Re s = 1/2. Riemann’s conjecture has been probed numerically but it is stillunsolved. In 1914, Hardy [107] proved that there are innitely many zeros of the zeta function onthe line Re s = 1/2. Later Selberg [197] proved a small positive fraction of zeros are on this line;this was improved by Levinson [142] to a third, and now thanks to Conrey [50] we know thatat least two-hs lie on the critical line. Assuming the Riemann hypothesis, Montgomery [160]rescaled in a suitable way the imaginary part of γ1 ≤ γ2 ≤ . . . of this zeros 1/2 + iγk and heobtained a limiting form for the pair distribution of these zeros. During a meeting in the 1970s,Montgomery showed his pair correlation function to Dyson who immediately recognized it asthe pair correlation function of the GUE ensemble. Some masterful numerical computations byOdlyzko [176] of the zeros of ζ(s) showed remarkable agreement with Montgomery’s conjecture.For more historical details, a beautiful review is [96]. e striking observation that the nontrivialzeros are distributed like the spectrum of the GUE ensemble in one of the most notable “applicationof RMT in mathematics”.

1.3.3 Concentration of measure for linear statistics

Suppose we have a bunch of scalar random variables X1, . . . , Xn. What can we say about theirsum Sn = X1 + · · ·+Xn? If each individual summand Xi has range in an interval of size O(1),then their sum Sn varies in an interval of sizeO(n). However, a remarkable phenomenon, knownas concentration of measure, asserts that if the summands Xi are statistically independent, thissum sharply concentrates in a much narrower range, typically of size O(

√n). is phenomenon

is quantied by a variety of large deviation inequalities that bound the probability that Sn deviatessignicantly from its mean. is concentration holds in a more general seing assuming a sucientamount of independence between the summands. e same phenomenon applies not only tolinear expressions such as Sn = X1 + · · ·+Xn but more generally to nonlinear combinations

16

F (X1, . . . , Xn) of such variables, provided that the nonlinear function F is suciently regular (inparticular, if it is Lipschitz). Do concentration of measure phenomena emerge in RMT? Perhaps,the most simple quantities one can investigate for a random matrix Mn are their linear statisticswhich are sum functions of their random eigenvalues F (λ1, . . . , λn) =

∑i a(λi). In Experiment

3 we will consider a very simple linear statistics, namely the trace of a positive random matrixmodel: F (λ1, . . . , λn) =

∑i λi.

Experiment 3 Concentration of linear statistics1: for k = 1, . . . , iter do2: Construct an n× n random matrix A as in Experiment 1;3: Mn = AA†/n; (∗ Construct the scaled positive matrix Mn ∗)4: λ = Eigenvalues[Mn];5: S

(k)n =

∑i∈[n] λi; (∗ Compute the trace of W ∗)

6: end for7: Histogram[Sn]; (∗ Take the histogram of Sn =

(S

(1)n , S

(2)n , S

(3)n , . . . , S

(iter)n

)∗)

8: Compute Mean[Sn] and Variance[Sn].

Again, one should perform Experiment 3 for increasing sizes n. Two natural questions are:

i) Can you guess the “shape” of the output Histogram[Sn] for large n?

ii) Can you guess the scaling of Mean[Sn] and Variance[Sn] with n?

A naive approach to question i) is to consider that, as n increases, we are taking the sum of alarge number of random variables λi = O(1). If one suspects the existence of a limiting law, arst candidate is the normal distribution, even if at this stage we do not have any informationabout the nature of the random variables λi’s. e output showed in Fig. 1.4 shows that acentral limit theorem holds and the limit distribution of

∑i∈[n] λi is a Gaussian. To complete the

characterization of the limiting law we should x the two parameters of the Gaussian law, that isto answer to question ii), and here comes the surprise. While Mean[Sn] = O(n) we see that theuctuations behave quite dierently: Variance[Sn] = O(1). In other word, the uctuations arefreezed, in the sense that they do not scale with n. is phenomenon holds for many dierentlinear statistics An(Mn) =

∑i∈[n] a(λi) with general hypotheses on a(x) (one could try with∑

i∈[n] λ2i ,∑

i∈[n] 1/λi,∑

i∈[n] expλi). In the i.i.d. scenario, the sum of n i.i.d. random variableshas mean O(n) and uctuations of order O(

√n). is could be taken as a signature that the

eigenvalues of a random matrix are strongly correlated random variables.

(i.i.d. random variables) Mean[Sn] = O(n) Variance[Sn] = O(n)

(eigenvalues of random matrices) Mean[Sn] = O(n) Variance[Sn] = O(1)

When a Gaussian behavior is expected, the job is to compute the average E[An] and the varianceVar(An) = E[A2

n]−E[An]2 of the linear statistics An. en, it is of main importance to have a

17

46 48 50 52 54Sn

n = 50

96 98 100 102 104Sn

n = 100

196 198 200 202 204Sn

n = 200

Figure 1.4: Histogram of the of Sn =∑i∈[n] λi (see Experiment 3) for dierent values of n. e solid red line is a

Gaussian density with mean E[Sn] = n and variance Var(Sn) = 2 (see equations (5.63) and (5.64) in Chapter 5). It isquite easy to see that the mean scales linearly with n while the mean square deviation is almost constant (independentof n).

clear and practical algorithm that provides the leading order behavior of average and varianceof a linear statistics of a random matrix in the large dimensional limit. How to compute E[An]and Var(An) for large n? Let us focus rstly on the average. Consider a sequence of hermitianrandom matrices Mn and a linear statistics An = tr a(·) well-dened on that ensemble. Whatis the large n behavior of E[An(Mn)]? Suppose that the ESd µMn converges in expectation to anonrandom limit measure µ. en

limn→∞

1

nE[tr an(Mn)] = lim

n→∞

1

nE[n 〈µMn , a〉] = 〈E[µMn ] , a〉 = 〈µ , a〉 . (1.27)

erefore, the large n behavior of EAn is completely specied by the large n behavior of EµMn :

EAn(Mn) = n 〈µ , a〉+ o(n) . (1.28)

In order to compute the variance Var(An) we need further information. We introduce the objectsthat are able to detect the uctuations around the mean value.

Denition 1.5. e two-points empirical spectral density (or 2-ESd) of a (possibly random) n-dimensional matrix Mn is

µ(2)Mn

=1

n(n− 1)

∑i,j∈[n]i 6=j

δλi(Mn) ⊗ δλj(Mn) ∈M(R× R) . (1.29)

e 2-ESd is normalized´R2 dµ

(2)Mn

(x, y) = 1 and it is easy to verify that

µ(2)Mn

= µMn ⊗ µMn + ν , ν(R2) = O(n−1) . (1.30)

In the large n limit, µ(2)Mn

µMn ⊗µMn so that it is customary to introduce a more rened object:

18

Denition 1.6. e connected correlation measure of a (possibly random) n-dimensional matrixMn is

µcMn:= µ

(2)Mn− µMn ⊗ µMn . (1.31)

Observe that µcMnis not a probability measure.

e variance of An is

Var(An) = Var(n 〈µMn , a〉] (1.32)= n2 (E[〈µMn , a〉 〈µMn , a〉]−E[〈µMn , a〉]E[〈µMn , a〉]) (1.33)= n2 (〈E[µMn ⊗ µMn ] , a⊗ a〉 − 〈E[µMn ]⊗E[µMn ] , a⊗ a〉) , (1.34)

and, in general, for any two linear statistics An = tr a(·) and Bn = tr b(·) their covariance isgiven by

Cov(An, Bn) = n2 〈E[µMn ⊗ µMn ]−E[µMn ]⊗E[µMn ] , a⊗ b〉 (1.35)

e idea is that the average of An (or Bn) is known once we know the average of µMn . eircovariance is known once we know the “covariance” of µMn :

E[An] = n〈EµMn , a〉 (1.36)Cov(An, Bn) = n2〈Cov(µMn , µMn), a⊗ b〉 (1.37)

Suppose now that not only the density of states µ = limEµMn exists, but also n2Cov(µMn , µMn)converges to a O(1) measure K on R2 (the so-called correlation kernel). en:

EAn ∼ n 〈µ , a〉 , (1.38)Cov(A,B) ∼ 〈K , a⊗ b〉 . (1.39)

We will see in the next chapters that both the density of states µ and the correlation kernel Kexist for many ensembles of random matrices. Surprisingly enough, in these cases the correlationkernel K exhibits a stronger universality than µ.

Multivariate statistics Multivariate statistics concerns mostly with inference from multivari-ate data. e denite references for this subject are the books of Wilks [231], Anderson [9],Muirhead [162] and Johnsson and Wichern [121]. e multidimensional scenario in quite com-mon in science when several variables (characters or variates or observables) are simultaneouslymeasured over a population sample. Suppose that an investigator collects the values of n variatesfrom a sample of size m. If we denote the value of the j-th variate (j ∈ [n]) relative to the i-thindividual of the sample (i ∈ [m]) by xij , the data can be conveniently stored in a m× n matrixX = (xij) whose j-th column ~xj = (xij)1≤i≤m is the data vector encoding the m observationsof the j-th observable. e mean of ~xj is given by

xj =1

m

∑i∈[m]

xij (1.40)

19

while the sample covariance matrix S is a n-dimensional matrix composed by the mean-correctedsecond moments

Sij =1

m− 1

∑k∈[m]

(xki − xi) (xkj − xj)

=1

m− 1

(X − X

)T (X − X ) . (1.41)

where X is the matrix whose j-th column is identically equal to xj . Generally one performsa pre-processing step, called “centering”, on the data matrix by substracting to each column~xj its average xj . is corresponds to the transformation X − X 7→ X so that the samplecovariance matrix for this centered data matrix is simply S ∝ X TX . e symmetric matrix Scontains n variances (on the diagonal) and n(n− 1)/2 potentially dierent covariances (cross-correlations) on the o-diagonal terms. From a theoretical perspective one needs some specicsabout the process that generates the data. e standard seing assumes that the data comefrom a multivariate Gaussian distribution. e multivariate normal distribution is the basicbuilding block of multivariate analysis. e rst reason relies on the empirical fact that generalmultivariate observations are approximatively normally distributed. A second reason is thetheoretical possibility of exploiting the nice properties of the Gaussian law that make manyproblems tractable, whereas other probability laws turn out to be not manageable. In particular itcan be shown that, under the normality hypothesis of the data set, the vector of the averages Xand the sample covariance matrix S are sucient statistics. It should be clear that any inferencetechnique, for instance a hypothesis testing, involves the distribution of objects that depend on thedata matrix X = xij . Wishart [237] rstly considered a Gaussian model for a multivariate dataset and computed the distribution of the associated sample covariance matrix S . is distribution,which is the natural extension of the chi-square distribution in dimension n > 1, is nowadaysreferred as the n-dimensional Wishart distribution with m degrees of freedom. is is the rst (orat least the ocial) appearance of random matrices in the scientic literature. We will show inChapter 5 that an understanding of the linear statistics of S is an indispensable tool to assess thenormality hypothesis of a data set.

Mesoscopic conductors Systems intermediate between the macroscopic and microscopicregimes are referred as mesoscopic. Roughly speaking, such a systems are small enough to requirea quantum mechanical treatment and suciently large to be amenable of a statistical approach.Nowadays this scenario is ordinarily implemented in electronic systems (and devices) thanks tothe technological progress in semiconductors science. e electronic transport in mesoscopicsystems can be signicantly dierent from its classical counterpart. e typical seing is a cavityetched in semiconductors and connected to the external world by two aached leads. e cavityis brought out of equilibrium by an applied external voltage. e two incoming leads may ingeneral support n1 and n2 (with n1 ≤ n2) open electronic channels, i.e. dierent wave numbersof the incoming electronic modes. e S matrix, having a natural block form

S =

(r t′

t r′

), (1.42)

20

in terms of reection (r, r′) and transmission (t, t′) sub-blocks, connects the incoming andoutgoing electronic wavefunctions. Conservation of electronic current implies that S is unitary.e Landauer-Buiker theory [136, 86, 115, 39, 40] expresses most physical observables in termsof the eigenvalues (λ1, . . . , λn1) of the Hermitian matrix tt†. In particular, the conductance canbe viewed as a scaering problem governed by S .

Assuming that the average electron dwell time (the time spent inside the cavity) is well inexcess of the ergodic time (the time necessary to the ergodic exploration of the phase space) atlow temperatures and applied voltage, statistical properties of the electronic transport through acavity exhibiting chaotic classical dynamics display remarkably universal features.

For instance, two remarkable features are universal conductance uctuations (UCF) and weaklocalization [23]. e discovery of the phenomenon of universal conductance uctuations indisordered metallic samples, pioneered by Altshuler [5] and Lee and Stone [140] has had aprofound impact on our current understanding of the mechanisms of quantum transport atlow temperatures and voltage. ere are two aspects of this universality, i) the variance of theconductance is of order (e2/h)2, independent of the sample size or the disorder strength, andii) this variance decreases by precisely a factor of two if time-reversal symmetry is broken by amagnetic eld. Both features, observed in several experiments and numerical simulations (see[19] for a review), naturally emerge from a random-matrix theoretical formulation [23] of theelectronic transport problem [116, 15]. e phenomenon of UCF is just, however, one of thevery many incarnations of a more general and intriguing property of sums of strongly correlatedrandom variables. In the RMT approach2, the scaering process inside the cavity is governedby a scaering matrix S drawn at random from the unitary group [15, 116]. Within this RMTprescription, the conductance and other experimentally accessible quantities can be expressed aslinear statistics of a certain matrix model. We will elaborate deeper on this model in Chapters 6and 7.

Entanglement distribution in large quantum systems A quite common approximation inphysics is that, if some phenomenon of interest is localized and far from the rest of the world it canbe described as if nothing else were present in the universe. is is the approximation of “closedsystem” (see [98] for a recent interesting essay). is approximation is familiar in both classicaland quantum-mechanical models. In quantum mechanics, the state of an isolated system S isspecied by a normalized vector |ψ〉 ∈H belonging to a complex separable Hilbert space H . Asa mathematical theory, quantum mechanics is a linear theory. If the system of interest S consistsof several parts, say two subsystems S = A ∪ B, we insist that the theory is linear wheneverwe observe processes pertaining to the individual subsystems. In this bipartite scenario, if wepretend to linearize a bilinear theory we should consider as state space the tensor product of thestate spaces HA and HB pertaing to subsystems A and B. is state space is the tensor productH = HA ⊗HB . A very beautiful and elegant introduction to tensor products in mathematicsand physics can be found in the book of Schwartz [195].

2Another approach in RMT considers directly a random model of the Hamiltonian of the system; we will notconsider this approach here.

21

For a pure states |ψ〉 ∈ HA ⊗HB it is not true in general that |ψ〉 = |ϕ〉 ⊗ |ω〉 for some|ϕ〉 ∈ HA and |ω〉 ∈ HB . is apparently innocuous observation is the simplest incarnationof the so-called entanglement, a word that stands roughly speaking for the purely quantum-mechanical correlations among quantum systems. ese quantum correlations play a key role inthe quantum information theory and the task of entanglement quantication is a very active eldnowadays. For pure states of bipartite system, it is possible to quantify these quantum correlationusing a family of “equivalent” entanglement measures [112].

In the last years, many researchers tried to answer the following question: what is the typicalentanglement of a pure state for a large quantum system? Once the question has been well-poseda lot of answer came from RMT [53]. Indeed, many entanglement measures for random purestates can be expressed as linear statistics on some random matrix ensembles (see Chapter 8).Asking what is the typical entanglement is then equivalent to ask what is the average value ofsome linear statistics and to show the concentration of measure about this mean value in thelarge dimensional limit.

Chapter 2

Spectral Analysis of LargeDimensional Random Matrices

We will mostly be concerned with self-adjoint matrices in which case all the eigenvalues arereal. A consequence of the perturbation theory of normal matrices [127] is that the set of theeigenvalues λi(Mn) of a self-adjoint matrix Mn is a continuous function of Mn (in particular,if Mn is a random matrix then the eigenvalues λi(Mn) are random variables). But thesefunctions have no closed form for a generic matrix when its dimension n is larger than 4. Whatif we redened the problem, replacing “generic” with “random”? Surprisingly, many times theproblem becomes easier and the larger is the size of the matrix the easier is the answer. So specialmethods are needed to understand these “formulae” for eigenvalues of large dimensional randommatrices. ere are four important methods employed in this area: the moment method, theStieltjes transform method, the saddle-point method and the tools of Free Probability. For instance,suppose that we want to prove the existence of the density of states (1.11) of a random matrixmodel Mn and nd it. What we are looking for is a probability measure such that

ˆdµMn(x)ϕ(x)→

ˆdµ(x)ϕ(x) (2.1)

for all continuous test functions ϕ vanishing at innity. In order to prove this vague convergence,it is sometimes sucient to prove (2.1) for a suciently “informative” family of test functions ϕ.Indeed, this is the spirit of the moment method and the Stieltjes transfom method:

• Moment method: prove (2.1) forϕ(x) = xk : k ∈ N

whose span is dense in the space

of continuous functions on any compact set. Indeed, if µ is compactly supported, this issucient to establish the vague convergence.

• Stieltjes transform method: prove (2.1) forϕ(x) = 1

z−x : Im z > 0

. By the conti-nuity eorem 2.1, by showing the convergence pointwise in z we can deduce the vagueconvergence of the ESd.

22

23

ese methods are very powerful in nding the limiting spectral distribution of a random matrixmodel and, combined with concentration of measure results, are oen used to prove the almostsure converges of the ESd. In particular, they are tailored for matrix models where the distributionof the entries is known (i.e. Wigner matrices).

On the other hand, the philosophy of the third method is quite dierent:

• Saddle-point method: nd the density of states as the (unique) solution of a variationalproblem.

e saddle-point method is applicable for the invariant ensembles whose law of the eigenvalues isknown. In these situations a blend of the saddle-point method and the Stieltjes transfom methodnaturally comes out.

In this Chapter we will also introduce some very basic concepts of Free Probability and itsrole in the large dimensional limit of random matrices.

• Free probability: random matrices are noncommutative random variables. eir momentsare expectation with respect to a (tracial) state. e large dimensional limit consists essen-tially in the large dimensional limit of the collection of moments. Independent randommatrices (as matrix-valued classical random variables) become, under some conditions,freely independent random variables (in the noncommutative sense).

Among the uses of the asymptotic freeness of random matrices are the study of the large di-mensional limit of random matrices with free probability techniques. ese techniques will beprecious in the last Chapter of this thesis.

2.1 Moment method

One of the most popular technique in RMT is the moment method (see for instance Chapter 2of [211], Chapter 2 of [8] and the explicit examples in [12]), which use the moment convergencetheorem. As noticed before, the k-th moment of the ESd of Mn can be wrien as

ˆdµMn(x)xk =

1

ntrMk

n . (2.2)

is expression plays a fundamental role in RMT . By applying the moment converge theorem [12],the problem of showing that the ESd’s of a sequence of random matrices Mn tend to a limitreduces to showing that, for each k, the sequence 1

ntrMkn tends to a limit ak and then verifying

Carleman’s condition: ∑k≥1

a−1/2k2k = +∞ . (2.3)

We note that, in most cases the limiting spectral density has nite support, and hence the necessarycondition (2.3) for the moment convergence theorem holds automatically. Most results in thespectral theory of large dimensional random matrices (proving the existence or nding the

24

density of states) were obtained by estimating the mean, variance or higher moments of 1ntrMk

n .Expanding the traces

1

nE[trMk] =

1

n

∑i1,...,ik∈[n]

E[Mi1i2Mi2i3Mi3i4Mi4i5 · · ·Mik−1ikMiki1 ] , (2.4)

we see that denitely we have to compute the expectation of “words” (with lenght k) fromthe alphabet An = Mij : 1 ≤ i ≤ j ≤ n whose leers are the random variables Mij . eessential steps of the moment method are the following

(i) Compute the limiting expectation of words limn→∞1nE[trMk], for all k ≥ 1. is is

usually realized by establishing a bijective correspondence between “words” and “diagrams”.Once this diagrammatic representation is at hand, the job is to identify the dominantcontribution among the words in the large n limit, and compute the contribution of theseleading diagrams with combinatorial techniques;

(ii) Denoting by ak = limn→∞1nE[trMk] the moments computed in the previous step, show

that the ak’s satisfy the Carleman’s condition (2.3);

(iii) Identify the probability measure µ whose moments are given by ak =´

dµ(x)xk. is isusally done by rst calculating the generating function of the sequence ak.

ese steps are sucient to prove the weak convergence ofµMn toµ in expectation (and sometimesin probability). In order to replace convergence in expectation with almost sure convergence,usually, similar combinatorial arguments are used to show

Var

(1

ntrW k

)= O(n1+δ) (2.5)

for some δ > 0. erefore, for every ε > 0, by Chebyshev’s inequality

∑n≥1

Pr

(∣∣∣∣ 1ntrW k − 1

nE[trW k]

∣∣∣∣ > ε

)≤∑n≥1

ε2

n1+δ< +∞ , (2.6)

and the almost sure convergence follows by the Borel-Cantelli lemma (all but a nite number ofsummands in the rst term of (2.6) are zero).

2.2 Stieltjes transform method

We now turn to another method, the Stieltjes transform method, which has a complex-analyticavour rather than the combinatorial one of the moment method. We have seen that the momentmethod is straightforward, but computationally demanding. Even if many results in RMT havebeen derived using the combinatorial and diagrammatic approach, the Stieltjes transform method

25

turns out to be more powerful. Indeed, this method returns the results on the ESd of randomHermitian models and is a more accurate tool to study the density of states at local scales [78].

Whereas the moment method relies on the identity (11.34), the Stieltjes transfom methodproceeds from the key identity

ˆdµMn(x)

1

z − x=

1

ntr (z −Mn)−1 , (2.7)

valid for any z /∈ suppµMn . We refer to the le-hand side as the Stieltjes transform of Mn or ofthe ESd µMn , and denote it by sMn or sn for short. By expanding in a neighborhood of innity,the Stieltjes transform can be viewed as a moment generating function

sn(z) =1

z+

1

n

∑k≥1

trMkn

zk+1. (2.8)

Given a generic probability measure µ on the real line, its Stieltjes transform is

sµ(z) :=

ˆdµ(x)

1

z − x, ∀ z /∈ suppµ . (2.9)

In particular sµ is well-dened on the open upper and lower half-planes of the complex plane.Some general but remarkable properties are

sµ(z)∗ = sµ(z∗) , (2.10)

and the pointwise bound|sµ(z)| ≤ 1

|Im z|. (2.11)

Similarly, from the normalization of µ and the dominated convergence, one has the crucialasymptotic

sµ(z) =1 + o(1)

z, (2.12)

for large z with |Re z|/|Im z| bounded (the rate of convergence depends on µ). It is easy to see,again by dominated convergence that

s′µ(z) = −ˆ

dµ(x)1

(z − x)2 . (2.13)

erefore sµ is analytic outside suppµ. e imaginary part of sµ(z) is particularly interestingfor our scopes. Writing z = λ+ iη, we observe that

Im sµ(z) =

ˆdµ(x)

η

(x− λ)2 + η2. (2.14)

is imaginary part can be conveniently expressed as a convolution

Im sµ(z) = −πµ ∗ Pη(λ) , (2.15)

26

where Pη is the Poisson kernel

Pη(x) =1

π

η

x2 + η2. (2.16)

It is known that kernels Pη , η > 0 form a family of approximations of the identity. us we have

− limη↓0

1

πIm sµ(λ+ iη)→ µ(−∞, λ) (2.17)

in the vague topology. e same result can be recovered by using a suitable contour integral inthe complex plane to have the equivalent formula

− limη↓0

sµ(λ+ iη)− sµ(λ− iη)

2πi→ µ(−∞, λ) . (2.18)

e following Stieltjes continuity theorem is the analogue of the continuity theorem for the charac-teristic function of scalar random variables. It plays a crucial role in the Stieltjes transfom method(as the moment convergence theorem does for the moment method).

eorem 2.1 (Stieltjes continuity theorem). Let µn be a sequence of random probability measureson the real line, and let µ be a nonrandom probability measure. en:

(i) µn vaguely converges almost surely to µ i sµn(z) converges almost surely to sµ(z) for everyz in the upper half-plane;

(ii) µn vaguely converges in probability to µ i sµn(z) converges in probability to sµ(z) for everyz in the upper half-plane;

(iii) µn vaguely converges in expectation to µ i E[sµn(z)] converges to sµ(z) for every z in theupper half-plane;

A complete proof can be found in [8] (eorem 2.4.4). e crucial identity in the Stieltjestrasform method is sn(z) = 1

ntr (z −M)−1. e main idea of the method is the precedessorcomparison: compare the Stieltjes transform sn(z) of the n × n matrix M with the transforms

(k)n−1(z) of the (n−1)× (n−1) le minorsM (k) obtained fromM with the k-th row and column

removed. e steps of the method are usually the following

(i) Show the stability of the Stieltjes transform sn(z) = sn−1(z) +O(n−1). is means thatevery single row and every single column individually have small inuence on the Stieltjestransform;

(ii) Use the Schur complement formula to write a recursive relation of sn(z) in terms of thetranssform of some of its (n− 1)× (n− 1) minors:

sn(z) =1

n

∑k∈[n]

1

Mkk − z + v†k(z −M (k)

)−1vk

, (2.19)

27

where vk is the k-th column of M with the k-th element removed (Mkk is k-th diagonalelement of M ). If the denominator Mkk − z + v†k

(z −M (k)

)−1vk can be showed to be

equal to g(z, sn(z)) + o(1), then the limiting Stieltjes transform s(z) is the xed point of athe map 1/g(z, s):

s =1

g(z, s). (2.20)

Various concentration of measure techniques could be used to complement this method andpromote the convergence to the almost sure sense.

2.3 Saddle-point method

e saddle-point method gives the large n leading contribution to matrix integrals. It relies heavilyon a 2D Coulomb gas analogy due to Dyson, who rstly recognized that the eigenvalues of somerandom matrix models behave as a Coulomb gas in a potential well [69]. is method is availablewhen one is able to integrate out the angular degrees of freedom of the matrix model and writethe matrix integral in terms of the joint probability measure of the eigenvalues. erefore, it isinapplicable whenever this joint law is unknown. In fact the saddle-point method is tailored tocompute matrix integrals of the form

ˆdMn f(Mn)e−βntrV (Mn) (2.21)

in the large n limit (the n in front of the potential ensures that the integral is O(1)). In (2.21) weare assuming V such that dMn e

−βntrV (Mn) is a nite measure and f integrable with respect tothat measure. e starting point of the variational method is eorem 1.2 whose “relevant” partis restated here for convenience: suppose that the law of X is

law(Mn) = const× e−βntrV (Mn)dMn (2.22)

with β = 1, 2 or 4 ifMn is real symmetric, complex Hermitian or quaternion self-dual, respectively.e law of the (real) eigenvalues λ = diag(λ1, . . . , λn) of Mn is

P(λ) =1

Zβ,n|∆(λ)|β e−βn

∑k V (λk) , (2.23)

Zβ,n =

ˆRn

dλ1 . . . dλn|∆(λ)|β e−βn∑k V (λk) , (2.24)

where ∆(λ) is the Vandermonde determinant. Formulae (2.23) and (2.24) have absolutely crucialconsequences for a detailed analysis of many statistical properties of n dimensional randommatrices when n→∞. e integrand in (2.24) can be recast in the following form

e−β[−∑i<j log|λi−λj |+n

∑k V (λk)] (2.25)

28

and then, we expect that the leading contribution of integrals with respect to P(λ) should comefrom from the n-uples at which the energy function

EV (λ) := −1

2

∑i 6=j

log|λi − λj |+ n∑k

V (λk) (2.26)

is minimum. In order to move to the large n limit, it is useful to introduce an “integral” version ofthe problem. For any n-uple λ = (λ1, . . . , λN ), if we introduce the normalized counting measureµλ on R

µλ =1

n

∑i∈[n]

δλi ,

ˆR

dµλ(x) = 1 , (2.27)

the energy function can be equivalently wrien as E(λ) = n2E [µλ] with:

EV [µλ] : = −1

2

¨x 6=y

dµλ(x)dµλ(y) log|x− y|+ˆ

dµλ(x)V (x) , (2.28)

=

¨dµλ(x)dµλ(y)LV (x, y) , (2.29)

withLV (x, y) =

1

2(V (x) + V (y)− log|x− y|1x 6=y) . (2.30)

With this trick, we li a minimization problem of a function in Rn to a the minimization problemon the spaceM(R) of probability measures on the line. In the large n limit, we are led to thefollowing variational problem:

infµ∈M(R)

EV [µ] = infµ∈M(R)

¨dµλ(x)dµλ(y)LV (x, y) . (2.31)

One can show (eorem 2.3) that the inmum (2.31) is actually aained at a unique µ?V . isminimizer µ?V is called equilibrium measure.

2.3.1 Coulomb gas analogy and Fekete points

We have seen that the joint law (2.23) can be wrien as

P(λ) =1

Zβ,ne−βEV (λ) , (2.32)

EV (λ) = −1

2

∑i 6=j

log|λi − λj |+ n∑k

V (λk) . (2.33)

In this form, it is evident that (2.23) is the Gibbs-Boltzmann weight of a system of identical particleson the line with logarithmic repulsive interaction and subject to the external potential nV atinverse temperature β > 0. Such a system is referred to as a 2D Coulomb gas (the logarithm,

29

being the fundamental solution of the Laplace equation in dimension 2, is the 2D electrostaticinteraction between point charges). e normalization constant Zβ,n acquires immediately themeaning of the partition function of the system. As already anticipated, for large n the leadingcontribution to expectation values with respect to (2.23) should come from the congurationsλ? that minimize the energy function E(λ). For any nite n, it is easy to see that the energyfunction E(λ) is convex in the Weyl chamber (λ1, . . . , λn) ∈ Rn : λ1 ≤ · · · ≤ λn and goesto innity on the boundary of this chamber. erefore, it must have a unique minimum at apoint λ? = (λ?1, . . . , λ

?n) in the Weyl chamber called n-Fekete set of points with respect to V (we

are always assuming V continuous). e points that minimize E(λ) are indeed the ones thatmaximize ∏

i<j

|λi − λj |e−nV (λi)e−nV (λj) . (2.34)

When the potential is constant, we are looking for a system of points λ1, . . . , λn on a regionR ofthe real line that maximizes the geometric mean of their distances. Non trivial potentials weightthese distances.Remark 2.1. It is easy to see that the set of Fekete point are “almost ideal” for Lagrange inter-polation. Indeed the Lagrange interpolation formula for a generic polynomial pn of degree nreads

e−nV (x)pn(x) =∑i∈[n]

e−nV (λi)pn(λi)Ln,i(x) , (2.35)

whereLn,i(x) =

∏j∈[n]j 6=i

(x− λj)e−nV (x)

(λi − λj)e−nV (λi). (2.36)

From the extremality property of the Fekete points, we have |Ln,i(x)| ≤ 1 for any x in the regionof interestR and therefore the Lagrange constant (a measure of “goodness” of the intepolation) isbounded

L = supx∈R

∑i∈[n]

|Ln,i(x)| ≤ n . (2.37)

For any xed n, if V ∈ C1, at the minimum we have∇E(λ?) = 0, which expands to becomethe set of conditions

−∑j∈[n]j 6=i

1

λ?i − λ?j+ nV ′(λ?i ) = 0 , ∀ i ∈ [n] . (2.38)

is set of equations is nothing but the stationarity condition (zero force) for the Coulomb gasto be in equilibrium. ese conditions can be tackled by using an ingenious method due toStieltjes [206], which deals with the electrostatic interpretation of the zeros of some families oforthogonal polynomials. Let us introduce the monic degree-n polynomial

P (x) :=n∏i=1

(x− λ?i ) . (2.39)

30

Dierentiating this polynomial we get

P ′(x) = P (x)∑i∈[n]

1

x− λ?ifor x 6= λ?i with i ∈ [n] , (2.40)

and then ∑j 6=i

1

x− λ?j=P ′(x)

P (x)− 1

x− λ?i=P ′(x)(x− λ?i )− P (x)

P (x)(x− λ?i ). (2.41)

en, assuming that λ?i 6= λ?j for i 6= j, the interaction term can be recovered using the Bernoulli-L’Hospital rule

limx→λ?i

∑j 6=i

1

x− λ?j= lim

x→λ?i

P ′(x)(x− λ?i )− P (x)

P (x)(x− λ?i )= lim

x→λ?i

P ′′′(x)(x− λ?i ) + P ′′(x)

P ′′(x)(x− λ?i ) + 2P ′(x), (2.42)

and nally one gets ∑j∈[n]j 6=i

1

λ?i − λ?j=

P ′′(λ?i )

2P ′(λ?i ). (2.43)

erefore the stationarity conditions (2.38) can be equivalently wrien as

P ′′(λ?i )− 2V ′(λ?i )P′(λ?i ) = 0 , ∀ i ∈ [n] . (2.44)

eorem 2.2 (Stieltjes). If

V ′(x) =A(x)

B(x), (2.45)

where A and B are polynomials with degA ≤ 1, degB ≤ 2 and degA + degB < 3, then themonic polynomial (2.39) satises the dierential equation

B(x)P ′′(x)− 2nA(x)P ′(x) + n2CP (x) = 0 (2.46)

with a suitable constant C .

Proof. From the stationarity condition (2.44) :

B(λ?i )P′′(λ?i )− 2A(λ?i )P

′(λ?i ) = 0 , ∀ i ∈ [n] . (2.47)

e function B(x)P ′′(x)− 2A(x)P ′(x) is a polynomial of degree less or equal to n, with rootsin λ?i , i ∈ [n]. erefore it is proportional to P . e constant C is determined from the coecientof the leading term xn (a consistent choice of C is not possible if degA = 1 and degB = 2simultaneously).

31

One can study the “density” of the zeros of P (x) by looking at the function

φ(x) := P (x)e−V (x) , (2.48)

which is solution (keeping the leading terms in n) of the eigenfunction equation

B(x)φ′′(x) + n2

[γ − A2(x)

B(x)

]φ(x) = 0

(γ = lim

n→∞C). (2.49)

In the next chapters we will oen use the following summation formula∑i∈[n]

∑j∈[n]j 6=i

h(λi)

λi − λj=

1

2

∑i,j∈[n]i 6=j

h(λi)− h(λj)

λi − λj. (2.50)

We oer three immediate consequences of this summation formula. First, choosing h(x) = 1 weget that the sum of the internal forces of the Coulomb gas vanishes∑

i∈[n]

∑j∈[n]j 6=i

1

λi − λj= 0 , (2.51)

according to the principles of mechanical systems. Second, using h(x) = x we see that the virialassociated to the internal forces ∑

i∈[n]

∑j∈[n]j 6=i

λiλi − λj

=n(n− 1)

2, (2.52)

is independent of the conguration of the Coulomb gas. ird, using h(x) = (z − x)−1 withz 6= λi, for all i ∈ [n]:

∑i∈[n]

1

z − λi

∑j∈[n]j 6=i

1

λi − λj=

1

2

∑i,j∈[n]i 6=j

1

z − λi1

z − λj=

1

2

(∑i

1

z − λi

)2

+O(n) . (2.53)

One can show that if λ? is a n-Fekete set of points, then its normalized empirical measure µλweakly converges to the equilibrium measure µλ? µ?V [61]. en, turning to a RMT perspective,for a linear statistics An(Mn) = tr a(Mn), with Mn from an invariant ensemble one has

E[An(Mn)] =∑i∈[n]

a(λ?i ) + o(1) , (2.54)

where λ?1, . . . , λ?n are the equilibrium congurations (the n-Fekete points) of the associatedCoulomb gas.

32

2.3.2 Variational problem

Let us consider the functional (2.28) with V : R→ R∪+∞ bounded from below and satisfyingthe growth condition

lim inf|x|→+∞

V (x)

log |x|> 1 . (2.55)

We quote here the crucial existence and uniqueness theorem whose proof can be found in [119,61, 8].

eorem 2.3 (Existence and uniqueness of the equilibrium measure). Let V bounded from belowand satisfying the growth condition (2.55). en, there exists a unique probability measure µ?V suchthat

infµ∈M(R)

EV [µ] = EV [µ?V ] . (2.56)

Moreover, µ?V has compact support.

We look now for the Euler-Lagrange equations for the variational problem

infµ∈M(R)

EV [µ] . (2.57)

eorem 2.4 (Variational Equations: weak form). e equilibrium measure µ?V satises the fol-lowing conditions: there exists a constant C ∈ R such that

(i)´

dν(x)ε(x) ≥ C , ∀ ν ∈M(R) with compact support with E(ν) <∞;

(ii) ε(x) = C , µ?V − a.s. .

whereε(x) = V (x)−

ˆdµ?V (y) log|x− y| . (2.58)

Remark 2.2. e eective energy of a single charge of the Coulomb gas is the sum of the potentialenergy and the interaction with the other charges of the gas: V (λi)−

∑j 6=i log |λi − λj |. For large

n, a charge located in x has electrostatic energy ∼ n2ε(x). At equilibrium it should be clear thatthis eective energy is the same for all the particles of the gas. en, V (λi)−

∑j 6=i log |λi − λj | =

C , for all i ∈ [n] for some constant C . is is the meaning of condition (i) in the variationalconditions. Moving to a continuous description we have paid the price that the condition holdsfor all the positions where a charge exists except for a negligible (measure zero) set. e physicalmeaning of the constant C is that of chemical potential. More precisely, the variational problemshould be implemented by a Lagrange multiplier C that ensures that the minimization problemis performed on the space of normalized measures. On the equilibrium measure, this auxiliaryparameter acquires the meaning of chemical potential and is constant (almost) everywhere onthe support of the equilibrium measure.

If we secretely know that the equilibrium measure is absolutely continuous dµ?V (x) =%?V (x)dx, then we can write a stronger version of the variational problem.

33

eorem 2.5 (Variational Equations: strong form). Suppose that the equilibrium measure dµ?V (x) =%?V (x)dx for some continuous density %?V . With the same notation of the previous theorem, thereexists a constant C ∈ R such that

(i) ε(x) ≥ C , ∀x ∈ R;

(ii) ε(x) = C , on x : %?V (x) > 0 .

We seek µ?V in the form dµ?V = %?V (x)dx, for some continuos %?V (x) ≥ 0 such that´dx %?V (x) = 1. If we indeed succeed in producing such a %?V , then %?V (x)dx is necessarily

the equilibrium measure. From eorem 2.5, the function ε(x) = V (x)−´

dy %(y) log|x− y|is constant function ε(x) = C on supp %?V . Outside the support of the equilibrium densityx 6= supp %?V the single particle energy ε(x) cannot be lower than C .Remark 2.3. We immediately see that possible solutions should have compact support (as guaran-teed by eorem 2.3). Indeed the equation ε(x) = C can be made more explicit as

V (x)− C =

ˆdy %(y) log|x− y| . (2.59)

and cannot be satised for |x| → ∞. Indeed the right-hand side of (2.59) is ∼ log|x| for large |x|and thus, by the growth condition (2.55), a dierent order with respect to the le-hand side.

A necessary condition for ε(x) to be constant on the support of %?V is the condition of vanishingderivative. We should compute the weak derivative of that function. Concerning the rst term

L(x) = −ˆ

dy %(y) log|x− y| (2.60)

we have

DL[φ] =

ˆdxφ′(x)

(ˆdy log|x− y|%(y)

)(2.61)

=1

2

ˆdxφ′(x)

(ˆdy lim

ε↓0log (|x− y|2 + ε2)%(y)

)(2.62)

=1

2limε↓0

ˆdxφ′(x)

(ˆdy log (|x− y|2 + ε2)%(y)

)(2.63)

=1

2limε↓0

ˆdxφ(x)

(ˆdy

2(x− y)

|x− y|2 + ε2%(y)

)(2.64)

=

ˆdxφ(x)

dy

%(y)

x− y(2.65)

whereffl

denote the Cauchy principal value.Remark 2.4. e integral version of the summation formula (2.50) is

ˆdx %(x)h(x)

dy

%(y)

x− y=

1

2

¨dxdy %(x)%(y)

h(x)− h(y)

x− y, (2.66)

34

that immediately implies the counterparts of (2.51),(2.52) and (2.53):ˆ

dx %(x)

dy

%(y)

x− y= 0 , (2.67)

ˆdx %(x)x

dy

%(y)

x− y=

1

2, (2.68)

ˆdx

%(x)

z − x

dy

%(y)

x− y=

1

2s2(z) ,

(s(z) =

ˆdx

%(x)

z − x, z /∈ supp %

). (2.69)

e stationarity condition ∂xε(x) = 0 reads explicitly

V ′(x) =

dy%?V (y)

x− y, (2.70)

for almost every point in x : %?V (x) > 0. We will refer to (2.70) as the saddle-point equation ofthe variational problem. e singular integral equation (2.70) is the integral counterpart of thestationarity conditions (2.38) and, in the electrostatic framework is the force balance conditionfor the density of %?V to be in equilibrium in the large n limit under the competing Coulombself-interaction and the external potential.

eorem 2.6 (Connected support). If V is convex, then supp %?V is connected.

Proof. Suppose that the support is the disjoint union of two intervals [a1, b1] ∪ [a2, b2] withb1 < a2 (this simplies the discussion but is not restrictive). From eorem 2.5 we should be ableto choose x1 ∈ [a1, b1], x2 ∈ [a2, b2] and α ∈ (0, 1) such that

ε(x1) = C (2.71)ε(x2) = C (2.72)

ε(αx1 + (1− α)x2) ≥ C . (2.73)

But, using the convexity of V and the strict convexity of the logarithmic interaction one has

ε(αx1 + (1− α)x2) < αε(x1) + (1− α)ε(x2) = C .

is contradiction establishes the lemma.

As the intuition suggests, when the potential V has a unique well the support of %?V has onlyone connected component supp %?V = [a, b]. In this case the solution is given explicitly by atheorem due to Tricomi [213]

eorem 2.7 (Tricomi). Let us consider the airfoil equation:

φ(x) =

b

ady

f(y)

x− y(2.74)

35

where φ belongs to a space Lp with p > 1, and a < b. Any solution of (2.74) belonging to Lp isgiven by the Tricomi formula:

f(x) =1

π√

(b− x)(x− a)

[K − 1

π

b

adx′

√(b− x′)(x′ − a)

x− x′φ(x′)

](2.75)

where K is the total mass of f :

K =

ˆ b

af(x)dx . (2.76)

If we know a priori that the solution %?V has a connected (and compact!) support, the theoremprovides the general form of %?V (x) = %?V (λ;K, a, b) with a single support supp %?V = [a, b]. Itdepends on three arbitrary constants that are determined by imposing the normalization conditionK =

´dx %?(x) = 1 and the behavior of %?V at the two edges a and b of the support. e condition

K = 1 can be immediately implemented to get the general form of a one-cut equilibrium measure

%?V (x) =1

π√

(b− x)(x− a)

[1− 1

π

b

a

√(b− x′)(x′ − a)

x− x′V ′(x′)dx′

]. (2.77)

We stress the fact that, since the equilibrium measure %?V is unique, there is only one “correct”choice of a and b in (2.77).

Multiple support solutions of the saddle-point equation escape from the Tricomi theorem.ey are more complicate and involve, in general, more arbitrary constants. It is possible to showthat a solution %?V of (2.70) whose support has m connected components supp %?V =

⋃mi=1[ai, bi]

has the general form %?V (x) = %?V (x; ai, bi, κii∈[m]). Now we have to impose the behaviorat the edges ai, bi (i ∈ [m]) and m extra constraints, i.e. the so-called lling fractions of the mcomponents of the density, e.g.

´ biai

dx %?V (x) = κi [80, 125] (this is equivalent to impose theoverall normalization and the lling fraction of m− 1 connected components). How to x theselling fractions?

If ϕ is a solution of (2.70), then the weak derivative of ε(x) is zero almost everywhere onsuppϕ. If suppϕ = [a, b] is connected this is sucient to conclude that ε(x) is constant foralmost every point in [a, b] and then to conclude that %?V = ϕ. If suppϕ is not connected, thecondition of zero derivative is not sucient. In fact a function with vanishing derivative isconstant on each connected component of its domain and in our case ∂xε(x) = 0 in the [ai, bi]for all i ∈ [m] implies

ε(x) = κi , on x ∈ [ai, bi] : ϕ > 0 , ∀ i ∈ [m] (2.78)

In this case, we should add the conditions κi = κj (i 6= j) in order to conclude that ϕ = %?V . esem− 1 equations (in addition to the normalization constraint) provide the additional constraintsthat single out the unique equilibrium measure. At equilibrium, the lling fraction κi is such thata small variation δκi does not change the energy EV at rst order. To implement this “equality ofchemical potentials” condition (in the physical terminology), we could require that, at equilibrium,

36

the work required to move a single charge from bi to ai+1 is null [125, 80]. A charge located at xexperiences a force equal to −∂xε(x). en, the total work needed to move a charge from bi toai+1 is

−ˆ ai+1

bi

dx ∂xε(x) . (2.79)

en, a sucient condition for ε(x) = C for all x ∈⋃mi=1[ai, bi] is

∂xε(x) = 0 if ai < x < bi , ∀ i ∈ [m]´ ai+1

bidx ∂xε(x) = 0 , ∀ i ∈ [m]

. (2.80)

Here we explain how to implement the zero work condition. To do this we should make adigression on the Stieltjes transform. Suppose that ϕ solves the saddle-point equation (2.70).is summable solution is real and nonnegative on some intervals [ai, bi] of the real axis andcan be analytically extended outside these intervals. Remarkably, for x /∈ [ai, bi], ϕ is purelyimaginary [125]. For a generic summable function ϕ(x) we dened its Stieltjes transform by

s(z) =

ˆdx

ϕ(x)

z − x. (2.81)

e stationarity condition (2.70) and the Stieltjes inversion formula (2.17) can be summarized aslimη↓0

Re [s(x+ iη)] = V ′(x) ,

limη↓0

Im [s(x+ iη)] = −πϕ(x) .. (2.82)

e above relation are closely related to the Plemelj formula limη↓01

x+iη = P.V. 1x − iπδ0 inpotential theory. We are looking for a positive function %(x) = Reϕ(x) and we are concerningwith the associated single particle energy

ε(x) = Re

[V (x)−

ˆdy ϕ(y) log |x− y|

]. (2.83)

e derivative of ε(x) reads

dε

dx(x) = V ′(x)− Re

[limη↓0

s(x+ iη)

](2.84)

= V ′(x)− Re

[limη↓0

(Re s(x+ iη) + i Im s(x+ iη)

](2.85)

= V ′(x)− Re[V ′(x)− iπϕ(x)

]. (2.86)

endε

dx(x) =

0 if x ∈ [ai, bi]

−iπϕ(x) otherwise .. (2.87)

37

erefore, if ϕ satises both the saddle-point equation and the zero work condition (2.88)V ′(x) =

ffldyϕ(y)

x−y if ai < x < bi , ∀ i ∈ [m]´ ai+1

bidxϕ(x) = 0 , ∀ i ∈ [m]

, (2.88)

we conclude that %?V (x) = Reϕ(x) is the equilibrium measure.

2.4 Free Probability

Free probability is the mathematical theory that beer captures the essential features of randommatrices as noncommutative random variables. oting its inventor D. Voiculescu [225], freeprobability can be described by the exact formula:

free probability theory = noncommutative probability theory + free independence . (2.89)

e main advantage of free probability as a noncommutative probability theory is that it enablesthe study of random matrices as stand-alone entities, without referring to the individual entries toget probabilistic-type results. e connection between free probability and random matrices staysin the phenomenon that, under certain conditions (like unitary invariance), free independenceoccurs asymptotically among large random matrices.

2.4.1 Basic denitions and results

Here we recall some basic concepts of free probability theory. We refer the reader to [226] fromwhere most of the following denitions are borrowed.

Denition 2.1 (Non-commutative probability). A random variable is an element a ∈ A, whereA is a unital C∗-algebra endowed with a state, i.e. a positive normalized functional ϕ : A → C.We call (A, ϕ) a non-commutative probability space. Hereaer the state ϕ will be a trace, i.e.ϕ(ab) = ϕ(ba).

Random matrices are non-commutative random variables with respect to the expectation:

ϕn(Mn) :=1

nE[trMn] , (2.90)

for a n× n normal random matrix, where E is the expectation over classical random variables.

Denition 2.2 (Free independence). Let (A, ϕ) be a non-commutative probability space. Afamily of unital subalgebras (Ai)i∈I is called a free family of subalgebras if

ϕ(a1 · · · an) = 0 (2.91)

whenever

• aj ∈ Ai(j) with i(j) 6= i(j + 1) (1 ≤ j ≤ n− 1) and

38

• ϕ(ai) = 0 for all i ∈ [n]

A family of subset (Ωi)i∈I is called free if the unital subalgebras that generates form a free familyof subalgebras. A family of random variables (ai)i∈I is said freely independent if the family ofsubsets (ai)i∈I is free.

Example 2.1 (Computation with free variables). Let a and b be free.

(i) Suppose that we have to compute ϕ(ab). If we center our random variables, according tothe denition of “freeness” we have

0 = ϕ ((a− ϕ(a))(b− ϕ(b)))

= ϕ(ab)− ϕ(a)ϕ(b)− ϕ(b)ϕ(a) + ϕ(a)ϕ(b)

= ϕ(ab)− ϕ(a)ϕ(b) .

ereforeϕ(ab) = ϕ(a)ϕ(b) , if a and b are free . (2.92)

(ii) In the same way, by centering and using the linearity of ϕ we have that for a and b free

ϕ(akb`) = ϕ(ak)ϕ(b`) , (2.93)

andϕ(ak1b`ak2) = ϕ(ak1+k2)ϕ(b`) . (2.94)

(iii) In the above example the rule of freeness leads to the same conclusion expected for classicallyindependent random variables. To see the “departure” of the free calculation from theclassical one we investigate the expectation ϕ(abab). Starting from

ϕ ((a− ϕ(a))(b− ϕ(b))(a− ϕ(a))(b− ϕ(b))) = 0

we getϕ(abab) = ϕ(a2)ϕ(b)2 + ϕ(a)2ϕ(b2)− ϕ(a)2ϕ(b)2 . (2.95)

e above expectation should be compared with

ϕ(a2b2) = ϕ(a2)ϕ(b2) 6= ϕ(abab) . (2.96)

is example shows that free random variables cannot commute in general. is meansthat freeness is a genuinely noncommutative concept.

(iv) e previous computation shows that a and b can commute and be free simultaneously ifand only if at least one of them has zero variance (e.g. ϕ((a− ϕ(a))2) = 0). As a specialcase, if a and b are classical random variables, then they can be free if and only if at leastone of them is almost surely constant.

39

Denition 2.3 (Joint distribution). If (ai)i∈I is a family of random variables in (A, ϕ), letC 〈Xi|i ∈ I〉 be the free unital algebra overCwith generators Xiı∈I and leth : C 〈Xi|i ∈ I〉 →A be the homomorphism such that h(Xi) = ai, i ∈ I . e joint distribution of the family (ai)i∈Iis the functional µ : C 〈Xi|i ∈ I〉 → C dened by µ = ϕ h.

e reader unfamiliar with the notion of free algebras could think at C 〈Xi|i ∈ I〉 as theset of non-commutative polynomials in the indeterminates Xi with complex coecients.

Denition 2.4 (Limit distributions and asymptotic freeness). For all n ∈ N let (ai,n)i∈I be afamily of random variables (indexed by I) in (An, ϕn) and µn their joint distribution. en, µis called the limit distribution of these families as n → ∞, if limn→∞ µn(Y ) = µ(Y ) for everyY ∈ C 〈Xi|i ∈ I〉. If Iss∈S is a partition of I , the family of subsets ((ai,n)i∈Is)s∈S is calledasymptotically free as n→∞ if the distributions µn of (ai,n)i∈I have a limit distribution µ andif the family of subsets ((Xi)i∈Is)s∈S is free in (C 〈Xi|i ∈ I〉 , µ).

For a single random variable a in (A, ϕ) its distribution is a linear map µa : C[X] → C. Ifa = a∗ is self-adjoint then µa extends to a compactly supported probability measure on R.

If (ai,n)i∈I is a family of free random variables in (A, ϕ), then the distribution of ap+aq , withp, q ∈ I (and more generally of any polynomial in the ai’s) depends only on the distributions ofthe ai’s. e operation that associates to the pair of free distributions (µaq , µap) the distributionµaq+ap of ap + aq is called free additive convolution and will be denoted as µaq µap := µaq+ap .We will be interested in the following R-transform introduced by Voiculescu.

Denition 2.5 (R-transform). e R-transform Rµ of a probability measure µ is dened by theequation

sµ

(Rµ(z) +

1

z

)= z , (2.97)

where sµ is the Stieltjes transform of µ.

e nice feature of the R-transform is that it linearizes the free convolution. Indeed, it isan analogue of the logarithm of the characteristic function (the Fourier transform) for classicalvariables in the sense of the following theorem.

eorem 2.8 (See [204]). For any pair (µ, ν) of freely independent random variables the followingholds:

Rµν = Rµ +Rν . (2.98)

An important class of noncommutative random variables is the following.

Denition 2.6 (Haar unitary). Let (A, ϕ) be a noncommutative probability space. An elementu ∈ A is called a Haar unitary if u is a unitary element inA (i.e. uu∗ = u∗u = 1) andϕ(uk) = δk0

for every k ∈ Z.

Denition 2.7 (m-Haar unitary). Le (A, ϕ) be a noncommutative probability space. An elementu ∈ A is called a m-Haar unitary if u is a unitary element in A, um = 1 and ϕ(uk) = 0 if k isnot a multiple of m.

40

It is quite simple to provide matrix models for Haar unitary (with respect to the normalizedtrace). For instance, let U be a random matrices distributed according to the Haar measure onU(n), the unitary group acting on Cn. e two-side invariance of the Haar measure amounts tosay that

E [f(U)] = E [f(V UW )] , for allV,W ∈ U(n) , (2.99)for every measurable functions f : U(n) → C (whenever the expectation exists). Specializing(2.99) to V and W diagonal or permutation matrices we can state the symmetry properties

E [f ] = E

[f

([ei(θi+ψj)Uij

]i,j∈[n]

)], for all θi, ψj ∈ R (2.100)

E [f ] = E[f([Uσ(i)π(j)

]i,j∈[n]

)], for allσ, π ∈ Sn . (2.101)

It is easy to see that E[

1ntrUk

]= 0 for k ∈ Z− 0. Indeed, using (2.100) we have

E

[1

ntrUk

]= eikθE

[1

ntrUk

], (2.102)

for every θ ∈ R. is gives the conclusion.is example can be immediately generalized to U = U1 ⊗ U2 ⊗ · · · ⊗ Uk with Ui ∈ U(ni)

independently drawn from the unitarily invariant laws on U(ni), i ∈ [k]. An example of m-Haarunitary is provided by equidistributed random permutation matrices P ∈ S(m)

n , where

S(m)n := σ ∈ Sn | all cycles of σ have length m . (2.103)

2.4.2 Free probability and Random Matrix eory

We quote here the milestone results on the free property of random matrices

eorem 2.9 (Voiculescu [226]). Let (Un,i)i∈I be independent n× n unitary matrices distributedaccording to the Haar measure on U(n) and Wn a deterministic n × n unitary matrix. en((Un,i, U

†n,i

)i∈I

, (Wn,W†n))

is a family of pairs asymptotically freely independent. Morever, the

limit distribution of Un,i is the uniform law on the unit circle z ∈ C : |z| = 1.

eorem 2.10 (Voiculescu [226]). For each n let λ1(n) ≤ λ2(n) ≤ · · · ≤ λn(n) and σ1(n) ≤σ2(n) ≤ · · · ≤ σn(n) in a compact interval and such that

1

n

∑i∈[n]

δλi(n) µ

1

n

∑i∈[n]

δσi(n) ν .

as n → ∞. Let An and Bn be independent self-adjoint operators uniformly distributed on the setsof matrices with spectrum λi and σi, respectively. en An and Bn are asymptotically freeindependent.

41

2.5 Levy’s metric and perturbation inequalities

In this section we shall quote some inequalities useful to bound the dierences of the empiricalspectral distributions in terms of quantities “easy to compute”. We quote here the denition ofLevy’s metric and some lemmas we will use in Part III.

Denition 2.8 (Levy’s distance [143]). Levy’s distance L(F,G) between two cumulative distri-bution functions F and G on the real line is

L(F,G) := inf ε > 0 : F (x− ε)− ε ≤ G(x) ≤ F (x+ ε) + ε . (2.104)

e following theorem claries the usefulness of this distance.

eorem 2.11. Convergence of measures in the Levy metric sense is equivalent to convergence indistribution. us L provides a metrization of the weak topology.

For our scopes the following lemmas will be precious. Essentially they provide a control ofthe ESD’s (and then also the ESd’s) of two “close” (random) matrices.

Lemma 2.12 (eorem A.45 of [12]). Let A and B two n × n Hermitian matrices. en, Levy’sdistance between the ESD’s of A and B is bounded by

L(FA, FB) ≤ ‖A−B‖ . (2.105)

Lemma 2.13 (Corollary A.42 of [12]). Let A and B two n ×m matrices. en, Levy’s distancebetween the ESD’s of AA† and BB† is bounded by

L(FAA† , FBB†)4 ≤ 2

n2

[tr (AA† +BB†)

] [tr (A−B)(A−B)†

]. (2.106)

2.6 Examples and Applications

In this last section we give an overall picture for some classical random matrix ensembles. Wewill consider Wigner-Gauss G, Wishart-LaguerreW , Jacobi J and Cauchy C random matrices.All these models are self-adjoint ensembles. Hereaer a real, complex or real quaternion standardGaussian variable is a random variables with Gaussian densities

1√2πe−x

2ij/2 ,

1

πe−|zij |

2

,

(2

πe−2|zij |2 ,

2

πe−2|wij |2

). (2.107)

for β = 1, 2 and 4 respectively (recall that a real quaternion number is specied by two complexnumbers (z, w)). e symbol † stands for transpose, Hermitian conjugate and symplectic conjugaterespectively.

42

Example 2.2 (e Wigner-Gauss ensemble). Let Y be a n × n random matrix with standardGaussian independent entries. Gauss-Wigner random matrices are dened as (compare withExperiment 1 in Section 1.3.1):

G =1√2n

(Y + Y†

). (2.108)

For β = 1, 2 and 4, this ensemble is known as Gaussian Orthogonal (GOE), Gaussian Unitary(GUE) and Gaussian Symplectic (GSE) ensemble, respectively. e distribution of the eigenvaluesof G is given by (2.23)-(2.24) with

V (x) =x2

4. (2.109)

e reader should be alerted that the variance of diagonal and o-diagonal entries of G areactually dierent. As a consequence of the symmetrization G =

(Y + Y†

)/√

2n, the entrieson the diagonal are of the form Gii = 2Y and on the o diagonal Gij = Y1 + Y2, with Y, Y1, Y2

Gaussian independent variables. is dierence is crucial, as it ensures that the weight functionin the exponential can be wrien with V (x) as in (2.109).

Since nV ′(x) = nx/2, one can apply Stieltjes theorem 2.2 to obtain the dierential equation

P ′′(x)− nxP ′(x) + n2P (x) = 0 . (2.110)

In other words the monic polynomial P (x) =∏i(x − λ?i ) is a multiple of the n-th Hermite

polynomial Hn(√n/2x) [209], where

Hn (x) = n!

dn/2e∑k=0

(−1)k(x)n−2k

k!(n− 2k)!, (2.111)

with dxe donoting the smallest integer larger than x. One can study the zeros of this polynomialby looking at the function

φ(x) := P (x)e−V (x) . (2.112)

Keeping the leading term in n, φ solves the eigenfunction equation

φ′′(x) + n2

(1− x2

4

)φ(x) = 0 . (2.113)

Comparing with the harmonic oscillator equation φ′′(x) + k2φ(x) = 0, which has solutionsφ = A cos(kx + θ) for positive frequencies k2 and exponentially decaying solutions for k2

negative, we are led (heuristically, even if this argument can be made rigorous) to concludethat φ is concentrated in a region where

(1− x2

4

)is positive (i.e. inside the interval [−2, 2])

and will oscillate with frequency O(n). en, we expect to nd a density of state (the limitingdistribution of the λ?i ) supported on [−2, 2] with a mean spacing between the eigenvalues of G oforder O(1/n). Indeed, the saddle-point equation (2.70) for the equilibrium measure is:

x

2=

dy

%(y)

x− y. (2.114)

43

Since the potential V (x) is convex, one can use the Tricomi formula (2.77), obtaining:

%G(x) =1

π√

(b− x)(x− a)

[1− 1

π

b

adx′

√(b− x′)(x′ − a)

x− x′x′

2

](2.115)

=1

π√

(b− x)(x− a)

[1−

(x2

2− a+ b

4x− (a− b)2

16

)](2.116)

Since we know from eorem 2.4 that the saddle-point density is compactly supported we imposethe regularity conditions %G(a) = %G(b) = 0 to x the free constants (the edges of the support).We obtain a = −2, b = 2 according to the previous discussion. e saddle-point density is theso-called Wigner’s semicircle law:

%G(λ) =1

2π

√4− λ2 1|λ|<2 . (2.117)

We stress the fact that the density of states is independent of β, as already predicted by theheuristic discussion on the n-Fekete points set. For completeness, we show how to obtain thesemicircle law using a xed-point equation for the Stieltjes transform. Starting from the saddlepoint equation (2.114), one can multiply both sides by 1

z−x with z /∈ supp % and integrate withrespect to the measure %(x)dx:

1

2

ˆdx %(x)

x

z − x=

ˆdx %(x)

1

z − x

dy

%(y)

x− y. (2.118)

Using the summation formula (2.50), the right-hand side is recognized to be s2(z)/2 (from (2.69))where s(z) is the Stieltjes transform of the measure %(x)dx. e le-hand side can be convenientlymanipulated using the identity x

z−x = zz−x − 1. Finally, we get the algebraic equation

s2(z)− zs(z) + 1 = 0 , (2.119)

which yields

s(z) =z ±√z2 − 4

2. (2.120)

To gure out what branch of the square root one has to use here, we recall the asymptotics(z) = (1 + o(1))z−1 for large z. In particular, we can gure out the asymptotics of (2.120) forlarge real z to conclude that

s(z) =z −√z2 − 4

2. (2.121)

In order to get the density of states of Wigner-Gauss matrices, we apply the inversion formularecovering, for |λ| < 2,

− s(λ+ iη)− s(λ− iη)

2πi→ 1

2π

√4− λ2 (2.122)

as η ↓ 0.

44

Example 2.3 (e Wishart-Laguerre ensemble). A Wishart-Laguerre random matrix of size nand parameter c ≥ 1 is dened as

W =1

nY†Y (2.123)

where Y is a rectangular m× n random matrix with standard Gaussian independent entries andm = cn, c ≥ 1. e random matrixW is positive. It is worth to remark that the m-dimensionalmatrix W = 1

nYY† has the same rank ofW and the same nonzero eigenvalues. e corresponding

potential for this ensemble is

V (x) =1

2[x− (c− 1) log x] . (2.124)

Applying the Stieltjes trick (eorem 2.2), the n-Fekete points set is given by the zeros of Psatisfying

xP ′′(x)− n [x− c+ 1]P ′(x) + n2P (x) = 0 . (2.125)

e solution of (2.125) is a multiple of the associated Laguerre polynomial (this explains thesecond name of the ensemble) P (x) = L

n(c−1)−1n (nx) [209], where

L(α−1)n (x) =

n∑k=0

(α+ n− 1

n− k

)(−x)k

k!. (2.126)

By introducing the function φ(x) = P (x)e−V (x) with the potential (2.124) we obtain the eigen-function equation (to leading order in n)

xφ′′(x) + n2

[1− (x− c+ 1)2

4x

]φ(x) = 0 . (2.127)

An heuristic discussion predicts that φ is concentrated in the interval [(1 −√c)2, (1 +

√c)2],

again with oscillations of frequency O(n). is heuristic is readily conrmed by the variationalproblem

infµ∈M([0,+∞))

¨dµλ(x)dµλ(y)LV (x, y) . (2.128)

Assuming that the minimizer is an absolutely continuous measure, we write the saddle-pointdensity

1

2− c− 1

2x=

dy

%(y)

x− y, supp % ⊂ [0,+∞) (2.129)

which is solved by the Tricomi formula (2.77):

%W(x) =1

π√

(b− x)(x− a)

[1− 1

2π

b

adx′

√(b− x′)(x′ − a)

x− x′

(1− c− 1

x′

)](2.130)

=1

π√

(b− x)(x− a)

[1−

(x

2− a+ b

4− c− 1

2

(1−√ab

x

))](2.131)

45

yielding as saddle-point density %c ≡ %W the celebrated Marcenko-Pastur law [154]

%c(λ) =

√(λ− λ−)(λ+ − λ)

2πλ1 λ−<λ<λ+ , λ± =

(1±√c)2

. (2.132)

e density of states is supported on an interval of the positive half-line. For c > 1, %c(λ) isbounded, while for c = 1 it acquires an integrable (square root) singularity at λ = 0.

With the free probability theory developed earlier, we can give a much simpler proof of thelimiting ESd based on the R-transform (2.97). We will present in detail this method (and also themoment method for the Wishart ensemble) in Chapter 11.

In Part III we will use the explicit expression of the Stieltjes transform of (2.132) (see [12, 8])

EsW(z) = Etr (z −W)−1

=

ˆdλ

%c(λ)

z − λ

=(c− 1)− z +

√(λ− − z)(λ+ − z)2z

. (2.133)

We will also use the moments of the Marcenko-Pastur law:

E[trW`] =

ˆdλ %W(λ)λ`

=∑s=1

cs−1Nar(`, s) (2.134)

where Nar(`, s) (1 ≤ s ≤ `) are the Narayana numbers

Nar(`, s) =1

s

(`

s− 1

)(`− 1

s− 1

), (2.135)

(see Problem 6.36. in [205] and [164, 207]). e link between Narayana numbers and moments ofWishart matrices is a longstanding known fact [12].

Example 2.4 (e Jacobi ensemble). From the Wishart ensemble one can easily construct anothervery interesting ensemble for applications. An n× n Jacobi matrix is concretely constructed fromtwo independent n-dimensional Wishart matrices of parameters c1, c2 ≥ 1 as:

J = (W1 +W2)−1W1 (2.136)

It is not dicult to see that 0 ≤ J ≤ 1 and it is convenient to parametrize the model in terms ofαi = ci − 1 ≥ 0, i = 1, 2. e potential of the model is given by

V (x) =1

2[α1 log x+ α2 log (1− x)] , (2.137)

46

and V ′(x) = 12

[(1−x)α1−xα2]x(1−x) . In this case, the Stieltjes eorem 2.2 predicts for the monic

polynomial the following dierential equation

x(1− x)P ′′(x)− n [(1− x)α1 − xα2]P ′(x) + n2

[α1 + α2 + 1− 1

n

]P (x) = 0 , (2.138)

whose solutions are Jacobi polynomials. Again it is useful to introduce the function φ(x) =P (x)e−V (x) that for large n satises:

x(1− x)φ′′(x) + n2

[(α1 + α2 + 1)− [(1− x)α1 − xα2]2

4x(1− x)

]φ(x) = 0 . (2.139)

It is not dicult to recognize that φ is concentrated in the interval whose edges are

λ± =

(√1 + α2 ±

√(α1 + 1)(α1 + α2 + 1)

α1 + α2 + 2

)2

. (2.140)

Solving the saddle-point equation

α1

2

1

x− α2

2

1

1− x=

dy

%(y)

x− y. (2.141)

one gets%J (λ) =

α1 + α2 + 2

2πλ(1− λ)

√(λ− λ−)(λ+ − λ) 1 λ−<λ<λ+ , (2.142)

with support dened by (2.140). For any α1,2 > 0 this density is bounded. When α1 = 0 (resp.α2 = 0) the density %J (λ) has a square-root singularity at λ = 0 (resp. λ = 1). For α1 = α2 = 0this density is the arcsine law

%arc(λ) =1

π√λ(1− λ)

1 0<λ<1 . (2.143)

Example 2.5 (e Cauchy ensemble). Finally, the Cauchy ensemble can be obtained by a Cayleytransform on unitary matrices U distributed according to the Haar measure on the n-unitarygroup:

C = i (1− U) (1 + U)−1 , (2.144)

valid for −1 /∈ σ(U). e potential of the ensemble is

V (x) = log√

1 + x2 . (2.145)

Comparing with the hypotheses of the existence and uniqueness eorem 2.3 of the equilibriummeasure, one sees that the growth condition (2.55) on the potential V (λ) is not satised. In thiscase the equilibrium measure exists, is absolutely continuous but has unbounded support:

%C(λ) =1

π

1

1 + λ2. (2.146)

47

Since the support of %C is not compact, this saddle-point density cannot be recovered from theTricomi formula. Maybe the simplest way to obtain (2.146) is to recognize that the eigenvaluesλk of C are related to the eigenvalues eiθk of U in (2.144) by

eiθk =1− iλk1 + iλk

. (2.147)

It is known [64] (or eorem 2.9) that the empirical density of the eigenphases θk of a randomunitary matrix distributed according to the Haar measure weakly converges to

dµU (θ) Unif(0, 2π) . (2.148)

Using the relation θk = − arctan(

2λk1−λ2k

)one immediately obtains (2.146).

e Cauchy ensemble exhibits many peculiar features not shared by other random matrixensembles. One of them is the following: the Cauchy ensemble has average empirical spectraldensity E [µCn ] given by (2.146) for any dimension n. is fact follows from the aforementionedrelation (2.144) with the circular ensembles of unitary matrices. Indeed, the average ESd E [µUn ](i.e. the one-body marginal of the eigenvalues of Un) is uniform (by symmetry). en, the claimfollows again from (2.147). Other interesting properties are: if C belongs to the Cauchy ensemble,then i) C−1 is again distributed according to the Cauchy ensemble and, ii) every sub-block obtainedfrom C by removing some rows and the corresponding columns again belongs to the Cauchyensemble. ese two important features were noted in [37].

Chapter 3

Notions of Large Deviations eory

Large deviations theory deals with probability of rare events. When we say “rare” we do notmean Pr(Xn ∈ A) ' 0, rather we mean events for which 1

nα log Pr(Xn ∈ A) is nite for large nand some α (independent on n). An example may help to x ideas.

Example 3.1 (Large deviations for i.i.d. random variables). Suppose that we are considering alarge supply of i.i.d. random variablesX1, X2, . . . with mean zero and nite nonzero variance. eclassical theorems of probability theory provide information about their sum Sn = X1 + · · ·+Xn

for large n. From the law of large number we know that Sn/n converges to its mean 0. Looking atsmaller scales, the central limit theorem states that Sn/

√n is asymptotically normally distributed.

In particular, we can write

Pr(Sn ≥ `

√n)∼ 1

2π

ˆ +∞

`dx e−

x2

2 . (3.1)

is estimate works uniformly for ` in any compact set. However, it is easy to see that theestimate (3.1) does not hold in general if we allow ` to depend on n. Indeed, suppose that|Xi| ≤ C for some constant. en, the probability that the sum Sn is larger than `

√n with

` > C√n is zero, and this obvious observation is not captured by the central limit theorem.

e Gaussian approximation describes uctuations ` = o(√n). ese are referred as “typical”

uctuations (or deviations). Larger uctuations (not described by the central limit theorem) arecalled rare events or large deviations.

Following Ellis [77], there are two approaches to investigate the exponential rate of a “rare”event. e rst one is by looking at appropriate Borel sets A such that

Pr(Xn ∈ A) ≈ e−anI(A) → 0 , (3.2)

for some sequence an of diverging constants and some number I(A) ≥ 0 independent on n.e second approach focuses on the probability that the random variable Xn is close to a numberx ∈ R. In this case we seek for a function I(x) such that

Pr(Xn ∈ [x, x+ dx]) ≈ e−anI(x)dx , (3.3)

48

49

for some nonegative function I(x). e asymptotic estimates (3.2) and (3.3) can be made compati-ble without much eort (at least heuristically). Indeed (3.3) follows from (3.2) if the Borel sets Ain (3.3) are of the form B(x, r) = y : |x− y| < r and we stipulate

I(x) = limr↓0

I(B(x, r)) . (3.4)

Conversely, from (3.3) we can estimate

Pr(Xn ∈ A) =

ˆA

Pr(Xn ∈ [x, x+ dx]) ≈ˆA

dx e−anI(x) . (3.5)

Using a Laplace approximation it is plausible that for large n the above integral is dominated bythe largest value of the integrand

Pr(Xn ∈ A) ≈ exp

[−an inf

x∈AI(x)

]. (3.6)

erefore, when suitable interpreted, both (3.2) and (3.3) assert a large deviation principle withrate function I .

e modern theory of large deviations has been developed in the 1960s and 1970s from theindependent works of Donsker and Varadhan [66], and Freidlin and Wentzell [95]. Prior to thatperiod many results on the propability of rare events were known, but there was no unied andgeneral framework. Cramer’s theorem [52] and Sanov’s theorem [191] constitute the fundamentallarge deviations results for i.i.d. variables. However, the rst large deviation calculation in thehistory was carried out by Boltzmann [30]. At that moment both statistical mechanics and largedeviations theory were born. In fact, the original link between large deviations theory andstatistical mechanics has been subject of numerous elaborations in literature. Among all of them itis worth mentioning the papers of Oono [177], Ellis [75, 76] and Lanford [137]. oting Touchee,large deviations theory and statistical mechanics “refer to dierent views of the same theory: onedirected at its mathematical applications - the other directed at its physical applications” [214].An important generalization of Cramer’s theorem was achieved with the results of Gartner [101]and Ellis [72]. is result is now known in the literature as the Gartner-Ellis theorem.

3.1 Basic denitions and theorems

e modern theory of large deviations is built on Polish spaces (i.e. complete separable metricspaces). A familiar example is the usual ambient space X = Rn. A class of Polish spaces arisingnaturally in applications is obtained by taking a Polish space X and then considering the set ofprobability measuresM(X ) on it.

Denition 3.1. Let I be a function mapping the Polish space X into [0,+∞]. e map I is calleda rate function if it is not identically +∞ and the sets x : I(x) ≤ K are compact, for everyK < +∞.

50

In particular a rate function is lower semicontinuous. In the following, for all Borel sets A of aPolish space we will use the notation I(A) = infx∈A I(x).

Denition 3.2. A sequence of probability measures Prn on a Polish space X satises a largedeviation principle with

(i) speed an (going o to innity with n), and

(ii) rate function I : X → [0,+∞],

if for all measurable sets A of X :

− I(A) ≤ lim infn→∞

1

anlog Prn(A) ≤ lim sup

n→∞

1

anlog Prn(A) ≤ −I(A) , (3.7)

(A and A are the interior and the closure of A respectively).

As usual we stretch the language and speak of “large deviation principle of the randomvariables Xn”, meaning the large deviation principle for law(Xn).Remark 3.1. It is tempting to replace (3.7) with the more intuitive stipulation

limn→∞

1

anlog Prn(A) = − inf

x∈AI(x) . (3.8)

However one can see that the above requirement is too strong to be useful. is descends fromthe fact that Prn Pr does not mean Prn(A) → Pr(A) for all A. Rather, by the Portmanteautheorem (see Section D.2 of [62]), it means that

lim infn→∞

Prn(C) ≤ Pr(C) , for allC closed , (3.9)

lim supn→∞

Prn(O) ≥ Pr(O) , for allO open . (3.10)

ere are a few properties and results that we need [216, 214].

(Uniqueness) e rate function of a large deviation principle is unique. Without this theusefulness of the rate function would be limited;

(Typical uctuations) If I is the rate function of a large deviation principle, then inf I = 0and I(x?) = 0 for some x?. If I has a unique minimizer, then the large deviation principleimplies a corresponding law of large number. If I is regular in x? with I ′′(x?) > 0 the largedeviation principle implies a corresponding central limit theorem;

(Gartner-Ellis theorem) A large deviation principle holds if

J(s) = − limn

1

anlogE[e−sanXn ] (3.11)

51

exists for all s ∈ R, is nite and dierentiable. In this case, the rate function I(x) is theLegendre-Fenchel transform of J(s):

I(x) = sups∈R

[J(s)− sx] . (3.12)

e function J(s) is known as (scaled) cumulant generating function (cumulant GF forshort) of Xn. Both the cumulant generating function J(s) and the rate function I(x) areusually called large deviation functions. A stronger version of the theorem includes thepossibility that J(s) is not dened on the whole line. If J(s) is dened on an intervalI , the Gartner-Ellis theorem is still true with the hypothesis the |J ′(s)| → +∞ when sapproaches the boundary of I (this is known as the steepness condition);

(Laplace principle) e Laplace principle, known is physics as saddle-point approximation, hasbeen extended in an abstract seing in large deviation theory by Varadhan [216]. Varadhan’stheorem states that if Xn satises a large deviation principle with speed an (going o toinnity with n) and rate function I , then for any bounded continuous function g

− limn→∞

logE[e−ang(Xn)] = infx

[g(x) + I(x)] . (3.13)

A stronger version of the theorem could be applied to unbounded functions and in particularto g(x) = sx. In such a case Varadhan’s theorem is a kind of converse statement of theGartner-Ellis theorem:

J(s) = infx

[sx+ I(x)] ; (3.14)

(Contraction principle) Let Y be a Polish space and g : X → Y be a measurable map. If theX -valued random variables (Xn) satisfy a large deviation principle with rate function I andg is continuous on x ∈ X : I(x) < +∞, then Yn = g(Xn) satisfy a large deviationprinciple with rate function

Ψ(y) = inf x : y=g(x)

I(x) . (3.15)

e milestones of large deviation theory are the theorem of Cramer [52] on the large deviation ofthe empirical average 1

n

∑i∈[n]Xi of a sequence Xi of i.i.d. random variables, and the theorem

of Sanov [191] on the large deviation principle of its empirical measure 1n

∑i∈[n] δXi . e central

result of large deviation theory in RMT is the following theorem.

eorem 3.1 (Large Deviation Principle for the ESd, [8]). Let λ = (λ1, . . . , λn) be a randomvector with law

P(λ) =1

Zβ,ne−βEV (λ) , (3.16)

EV (λ) = −1

2

∑i 6=j

log|λi − λj |+ n∑k

V (λk) . (3.17)

Under the same hypothesis of eorem 2.3, the ESd µn satises a large deviation principle with

52

(i) speed an = βn2;

(ii) rate function I[µ] := EV [µ]− EV [µ?V ].

In this thesis we combine this theorem with the contraction principle to infere a large deviationprinciple for linear statistics on random matrices.

Corollary 3.2 (Large Deviation Principle for linear statistics). Let Mn be an invariant ensemblewith law

law(Mn) = const× e−βntrV (Mn)dMn (3.18)

where V is an extended, continuous function satisfying the growth condition (2.55). Let us denoteI[µ] := EV [µ]− EV [µ?V ] and An = tr a(Mn) a linear statistics. If A[µ] := 〈µ , a〉 is continous on µ : I[µ] <∞, then A = n−1An satises a large deviation principle with speed βn2 and ratefunction

ΨA(x) = inf I[µ] : A[µ] = x . (3.19)

In the next Chapter we will use the above corollary to establish a large deviation principle forsome relevant linear statistics on classical ensembles, and we will use the Laplace principle tond the large deviation functions.

Part II

THE COULOMB GAS METHOD

53

Chapter 4

e Coulomb gas method

e Coulomb gas picture readily explains some theorems for invariant ensembles of randommatrices, and is a useful device for making reasonable guesses on the behavior of such ensembles.For invariant matrices Xn, the eigenvectors are trivially distributed, and the statistical propertiesof random matrices are associated to the statistical mechanics of an underlying 2D Coulomb gassystem, the “gas” of eigenvalues. One of the problem inRMT is connected with methods to computeaverages. In the Coulomb gas picture this corresponds to compute average of thermodynamicalquantities. In the majority of cases, these averages are hard to compute for nite n. Usuallythe formulas that one can derive are complicated and not easy to survey. It is quite natural,therefore, to look for simpler and more convenient approximations for these averages. isproblem is overcome in statistical mechanics by looking for asymptotic formulas valid whenthe number of particles increases beyond any limit. In the thermodynamic limit the computationof the averages are oen replaced by the more doable determination of most probable valuesassumed to be approximately equal to the corresponding averages. is is exactly the spiritof the saddle-point method we discussed so far. Once the Coulomb gas system associated to amatrix model is identied, one computes the limiting empirical spectral density as the one thatminimizes the energy functional. e limiting ESd is the conguration of the Coulomb gas withminimal energy or, equivalently, the most probable conguration. For large n, this equilibriumdensity of the gas becomes more and more indistinguishable from the density of states (the largen limit average ESd). A second ingredient in statistical mechanics is the concept of self-averagingquantities. e huge number of congurations of a large system looks at rst a worrisome scenario.On the other hand, oen the uctuations of a given quantity around its average value becomenegligible as the number of particles increases. is phenomenon can be understood in terms ofthe limit theorems of probability theory. When the number of particle increases, a concentrationof measure phenomenon makes the thermodynamical quantities close to their averages, in thesense that the deviations from these averages are exponentially suppressed: oen a most probablevalue is also the typical value. In fact, a renement of the saddle-point method, establishes alarge deviation principle for the congurations of the Coulomb gas (eorem 3.1). Informally,the assertion is that the probability that a 2D Coulomb gas (the eigenvalues) is far from the

54

55

equilibrium conguration is exponentially suppressed when the number of particle increases.It is remarkable that for a system of n particles, the rate for this suppression is n2 (the speedof the large deviation principle). e precise measure of unlikeliness of an out-of-equilibriumconguration is exactly given by the energy cost of that conguration (see eorem 3.1). en, bythe Coulomb gas picture one computes the typical conguration of the Coulomb gas. is turnsout to be the typical ESd of the matrix model, i.e. the density of states.

In the following section we elaborate on some results on the Coulomb gas method (for linearstatistics). e interest of this section is largely methodological. With the statistical mechanicsideas in mind, we will prove that the standard Coulomb gas method can be suitably shortenedusing some thermodynamical identities. A tremendous amount of disconnected calculations canbe saved by exploiting these asymptotic relations. In the next Chapters we will provide someapplications.

4.1 Historical remarks

Here we explain how to use the so-called Coulomb gas method to establish large deviation formulasfor linear statistics on random matrices. e seminal idea of the Coulomb gas picture goes back tothe works of Wigner [234] in the late 1950s and Dyson [69] in the early 1960s. ey remarked thatthere is an exact correspondence between the eigenvalue distributions of some random matrixmodels and the statistical mechanics of a classical 2D Coulomb gas. e inverse temperature of thegas β is now known as the Dyson index and is equal to β = 1, 2 o 4 for real, complex or quaternionmatrices respectively. Recently, Edelman and Dumitriu [71] have constructed explicit matrixmodels (tridiagonal models) with arbitrary non quantized inverse temperatures β > 0. One of theconsequences of the correspondence found by Dyson, is that the thermodynamic notions can betransferred from the Coulomb gas to the eigenvalues series, providing a well-understood languageto discuss the spectral statistics of random matrices. In particular the Coulomb gas picture is aconvenient framework to investigate large dimensional random matrices; in the thermodynamicallimit, a Coulomb gas with an arbitrary large number of particles is “equivalent” to the series ofeigenvalues of an arbitrary large random matrix. e idea of the original Coulomb gas method isthat the probability law of a certain observable Xn (that depends on the eigenvalues of a n× ninvariant random matrix ensemble) can be expressed as a ratio of two partition functions

Pr(Xn ∈ [x, x+ dx]) =Zn (Coulomb gas constrained to haveXn = x)

Zn (unconstrained Coulomb gas)(4.1)

Dyson made the assumption that for large n the Coulomb gas obeys the laws of classical thermo-dynamics. He obtained many results based on the well-established statistical mechanics traditioninstead of arguments of a mathematically rigorous kind. Later the mathematics of the problemhas been investigated deeper thanks to the development of the modern ideas of large deviationtheory.

Recently the seminal ideas of Dyson have been rediscovered and pushed further to computethe large deviation functions of linear statistics and the extreme eigenvalues of some random

56

matrix models with a lot of applications [60, 220, 150, 63, 222, 223, 152, 81, 153, 155, 46]. However,a critical revision of these recent works, reveals that the full arsenal provided by statisticalmechanical arguments (that were the ground-eld of Dyson’s calculations) has been almostcompletely ignored or at least “forgoen”. ese statistical mechanics considerations have beenevolved to theorems in large deviation theory, and we will try to apply them to rediscover thevarious disconnected statements sparse in the physical literature.

4.2 Large deviation functions for linear statistics

In this section we consider n random variables λ1, . . . , λn jointly distributed according to the law

P(λ) =1

Zne−β[−

∑i<j log|λi−λj |+n

∑k V (λk)] , (4.2)

with β > 0, and where the partition function Zn (we have suppressed the β dependence forconvenience) ensures the normalization

´dλP(λ) = 1.

We have seen that the joint law of the eigenvalues λ1, . . . , λn of an invariant random matrixwith law(Mn) = const× e−βnV (Mn) can be explicitly wrien as (4.2), with a suitable potentialV and β = 1, 2 or 4 according to the underlying eld (R, C, or H). In view of certain applicationsin physics, we will study the general model (4.2). However, it is useful to have in mind a matrixmodel. Some concrete representations have been discussed in Section 2.6.Remark 4.1. e support of the joint law P(λ) will be in general a subset Λn (where Λ ⊂ R) ofRn. In this case we will stipulate that V (λ) = +∞ for λ /∈ Λ. e empirical spectral density(ESd) µn = 1

n

∑i δλi satises suppµn ⊂ Λ for all n.

Recall that a linear statistics of a random matrix model An = tr a(Mn) corresponds to a sumfunction of the λi’s:

An =∑i∈[n]

a(λi) (4.3)

= n

ˆdµn(λ) a(λ) . (4.4)

For the large dimensional analysis we introduced the energy functional

E [µ] = −1

2

ˆdµ(λ)dµ(λ′) log|λ− λ′|+

ˆdµ(λ)V (λ) , (4.5)

whose minimizer µ?V is called equilibrium measure. Moreover, the functional

I[µ] := E [µ]− E [µ?V ] (4.6)

is the rate function of the ESd of the Coulomb gas with speed βn2, i.e. for every Borel set A ofM(R)

− infµ∈A

I[µ] ≤ lim inf1

βn2log Prn(A) ≤ lim sup

1

βn2log Prn(A) ≤ − inf

µ∈AI[µ] . (4.7)

57

To avoid unsightly formulas involving a spurious β dependence we have suitably scaled the speedto make the rate function “independent” of β. Informally, this is the assertion that

Pr[µn] ≈ e−βn2I[µn] , (4.8)

where “≈” stands for logarithmic equivalence. e large deviation principle is extremely important.e following consequences may give an idea of its amazing power and may also serve as aremainder that the technicalities required to dene in a proper way a large deviation principlemust not be allowed to obscure the much wider scope of the thoery developed.

From now on, we shall assume true the hypotheses of eorem 2.3 in Section 2.3.2, and we willassume that the (unique) equilibrium measure is absolutely continuous dµ?V (λ) = %?(λ)dλ. epositive, compactly supported and normalized function %? (that depends on V ) is called saddle-point density or density of states and is the (positive and normalized) solution of the singularintegral equation

V ′(λ) =

b

a

%?(λ′)

λ− λ′, [a, b] ⊂ Λ , (4.9)

for almost every point λ : %?(λ) > 0 where a charge exists. In the following, for every linearstatistics An =

∑i a(λi) we will denote

A[µn] = 〈µn , a〉 ≡1

n

∑i∈[n]

a(λi) , (4.10)

where µn is the counting measure of the λi’s (the ESd of Mn for a random matrix). Observethat if An = O(n) then A = O(1) (although we have eliminated the subscript, there is still adependence on n). e contraction principle states that, under reasonable hypotheses, A inheritsa large deviation principle with the same speed and rate function encoded in I[µ]. For any n letus denote the probability density function of A = A[µn] by Pn(x). We write formally

Pn(x) = En[δ(A− x)] , (4.11)

where the expectation is with respect the joint law (4.2). We would like to nd the explicit formof the rate function Ψ(x) = inf I[µ] : A[µ] = x that satistifes a large deviation formula ofthe type

Pn(x) ≈ e−βn2Ψ(x) , (4.12)

where the notation is suggested by (4.8). e idea is to pass through the Gartner-Ellis theoremand the Laplace principle. Indeed with a suitable choice of variables the Laplace transform of(4.11) is

Pn(s) = En[e−βn2sA] , (4.13)

and the Laplace principle (Varadhan’s theorem) predicts that

Pn(s) ≈ e−βn2J(s) , (4.14)

58

where the cumulant generating function (GF) is the Legendre-Fenchel transform of the rate functionJ(s) = supx[−sx−Ψ(x)]. en, the job is to compute the cumulant GF and, if possible, deducethe rate function. If J(s) is dierentiable in s = 0, then the cumulants κ`(A) of A read

κ`(A) = βn2

(− 1

βn2

)`J (`)(0) , (4.15)

and in terms of the extensive variable An:

κ`(An) = βn2

(− 1

βn

)`J (`)(0) , (4.16)

where J (`)(0) is the `-th derivative of J(s) evaluated in s = 0. How to compute J(s)? is is thetask of the Coulomb gas method. e Laplace transform Pn(s) = En[e−βn

2sA] reads

Pn(s) =1

Zn

ˆdλe−β[−

∑i<j log|λi−λj |+n

∑k V (λk)+ns

∑k a(λk)] (4.17)

=Zn(s)

Zn(0)(4.18)

a ratio of two partition functions (Zn(0) ≡ Zn). e saddle-point method reveals that asymptoti-cally in n

Zn(s) ≈ e−βn2Es[%?s ] , (4.19)

where

Es[µ] : = −1

2

ˆdµ(λ)dµ(λ′) log|λ− λ′|+

ˆdµ(λ) [V (λ) + sa(λ)]

= E0[µ] + sA[µ] , (4.20)

and infµ Es[µ] = Es[%?s]. Here we are assuming that V + sa satises the hypotheses of eorem2.3, and that the minimizers admit a density. Comparing with the denition of J(s) we nallynd

J(s) = − 1

βn2limn→∞

logZn(s)

Zn(0)= Es[%?s]− E0[%?0] , (4.21)

and by the Gartner-Ellis theorem (provided some regularity on J(s) is obtained)

−Ψ(x) = infs

[J(s)− sx] (4.22)

= infs

[Es[%?s]− E0[%?0]− sx] (4.23)

= infs

[inf%Es[%]− E0[%?0]− sx

](4.24)

= infs

inf%

[E0[%]− E0[%?0]− s (x−A[%])] . (4.25)

59

On the other hand, the rate function derives directly from the large deviation principle of the ESdand the contraction principle

Ψ(x) = inf%E0[%]− E0[%?0] : A[%] = x , (4.26)

where the inmum is taken over the measure with a prescribe value´

dλ %(λ)a(λ) = x. Wesee that the formulation in the Laplace space (4.25), is nothing but the implementation of theconstrained minimization problem (4.26) with a Lagrange multiplier s (the Laplace variable) thattakes into account the constraint A[%] = x. en the minimizer in (4.26) solves the followingsaddle-point equations: [

V ′(λ) + sa′(λ)]

=

%?s(λ

′)

λ− λ′, (4.27)

ˆdλ %?s(λ)a(λ) = x . (4.28)

Classical route e classical version of the Coulomb gas method to establish the large deviationformulae (4.12) and (4.13) proceeds as follows

(i) Assuming that the mimizer of µ?s (4.20) has a density dµ?s = %?s(λ)dλ, solve the integralequation [

V ′(λ) + sa′(λ)]

=

%?s(λ

′)

λ− λ′, (4.29)

to get %?s . For each value of s we get the minimizer %?s of Es[·];

(ii) Compute the “energy” Es of the saddle-point densities by plugging %?s into (4.20);

(iii) Compute the cumulant GF as energy dierence:

J(s) = Es [%?s]− E0 [%?0] ; (4.30)

(iv) Compute Ψ(x) as Legendre trasform of J(s) from (4.25).

A critical revision of this method, reveals that one can introduce a shortcut based on a standardthermodynamical argument. Indeed, the probability law ofA and its Laplace transform are relatedby

Pn(s) = En[e−βn2sA] (4.31)

=

ˆdxPn(x)e−βn

2sx (4.32)

≈ e−βn2 infx[Ψ(x)+sx] . (4.33)

If both Ψ and −J are dierentiable and strictly convex, then the Legendre-Fenchel problem issolved by J ′(s) = x?(s) and Ψ′(x) = −s?(x). What is the meaning of these quantities? e

60

answer is provided by the saddle-point equations (4.27)-(4.28). Indeed, using the stationaritycondition of %?s , we see that

x?(s) = J ′(s) =d

dsEs[%?s] (4.34)

=δEsδ%

[%?s]∂%?s∂s

+∂Es∂s

[%?s] (4.35)

=

ˆdλ %?s(λ)a(λ) (4.36)

= A[%?s] . (4.37)

e function s?(x) is the inverse of x?(s), i.e. s?(x?(s)) = s.e identity (4.25) can be wrien in the (almost) symmetric form

J(s)−Ψ(x) = sx . (4.38)

is equation should be read carefully. Indeed, in (4.38), there is only one independent variable:either s or x. Indeed, if for instance J(s) and Ψ(x) are both regular, the relation between theconjugate variables x and s is x = x?(s) = J ′(s) or s = s?(x) = −Ψ′(x), and (4.38) should beintended either as J(s?(x))−Ψ(x) = s?(x)x or as J(x)−Ψ(x?(s)) = sx?(s). In any case wecan stipulate that

dJ(s) = x?(s)ds , with J(0) = 0 , (4.39)−dΨ(x) = s?(x)dx , with Ψ(x?(0)) = 0 . (4.40)

ese equalities show that, when a large parameter (n in this case) is involved, the Laplacetransform and the Legendre transformation are intimately connected through the saddle-pointapproximation of the thermodynamic limit.

Shortened route e thermodynamical relations (4.38) and (4.39)-(4.40) provide an alternativemethod to compute the large deviation formulae (4.12) and (4.13). e steps of this shortenedmethod are

(i) Solve the saddle-point equation (4.27) to get %?s ;

(ii) Make explicit the relation between x and s using the prescription of (4.28) by evaluatingx?(s) = A[%?s];

(iii) Compute the large deviation functions J(s) and Ψ(x) using the relations dJA(s) = x?(s)dsand dΨA(x) = −s?(x)dx (where s?(x) is the inverse of x?(s)). For instance, the cumulantGF is

J(s) =

ˆ s

0x?(s′)ds′ , (4.41)

where we have used J(0) = 0. A similar integral formula holds for Ψ(x).

61

e boleneck of the old method is the evaluation of the “energy” functional (4.20) at the saddle-point density, which entails the evaluation of many integrals, while the method proposed hereonly requires the explicit relation between the real variable x and the Laplace variable s from(4.38).

Besides providing an elegant shortcut with respect to the traditional method, this improvementcan be immediately generalized (and easily applied) to the study of joint linear statistics. Indeed,let us consider for instance two linear statistics An =

∑i a(λi) and Bn =

∑i b(λi) with a and b

regular enough to apply the contraction principle. e joint probability law of A = n−1An andB = n−1Bn and their joint Laplace transform reads

Pn(x, y) = E [δ (A− x) δ (B − y)] , (4.42)Pn(s, w) = E

[exp

−βn2 (sA+ wB)

], (4.43)

where the averages are again with respect to (4.2). We would like to establish large deviationformulas of the type

Pn(x, y) ≈ e−βn2Ψ(x,y) , (4.44)

Pn(s, w) ≈ e−βn2J(s,w) , (4.45)

where Ψ(x, y) and J(s, w) are the joint rate function and cumulant GF of the random variables Aand B. An almost equal reasoning as before lead us to stipulate the following dierential relation

dJ(s, w) = A[%?s,w]ds+B[%?s,w]dw , J(0, 0) = 0 , (4.46)

where %?s,w is the equilibrium measure of the following energy functional

Es,w[µ] = −ˆ

dµ(λ)dµ(λ′) log|λ− λ′|+ˆ

dµ(λ) [V (λ) + sa(λ) + wb(λ)] . (4.47)

e extension to more than two linear statistics is straightforward.Remark 4.2. Coming back to the single statistics discussion for simplicity, one oen nds thatthe cumulant GF J(s) is not analytic. e points where J(s) is not analytic correspond to phasetransitions of the associated Coulomb gas system and thus they are called critical points scr instatistical physics. e order of the phase transition at scr is dened as the smallest integer ` suchthat J (`) = d`J/ds` is discontinuous at scr. e previous discussion (4.38) shows clearly that theCoulomb gas undergoes a `-th order phase transition at scr if d`−1x?/ds`−1 is discontinuous atscr. is facilitates the characterization of the phase transition.

Chapter 5

Large deviations of spread measuresfor Gaussian data matrices

e complexity of many phenomena requires to collect data that include simultaneous measure-ments on several variables. e body of methodology to deal with multivariate data is calledmultivariate analysis. Multivariate methods have been regularly applied to problems arising inthe physical, medical and social science. Large part of these methods is concerned with statisticalinference, that is, reaching valid conclusions on the basis of sample information. In the multivariaterealm more mathematical tools and techniques are required compared to the univariate seing.In fact, the earliest interest in random matrices arose in the context of multivariate statistics withthe works of Wishart [237], Fisher [87] and Hsu [113]. However, the interaction between RMTand multivariate analysis has been really exploited quite recently. Nowadays, more and morevast data collections are available and oen the number of variates n is large and comparablewith the sample size m. Roughly speaking, we are interested simultaneously in many variables(the variates) of our population. e standard methods of mathematical statistics usually requirea sample size much larger than the number of observables m n. However, if the numberof variables n is truly large, the required sample sizes m n are simply unaainable in mostsituations. is scenario, on the other hand, provides a seing where the techniques of RMT areeectively useful.

For more details on the classical methods in multivariate analysis see for instance the classicalbooks of Wilks [231], Muirhead [162] and Johnsson and Wichern [121]. An excellent review onthe applications of RMT in multivariate statistics is [122].

5.1 Introduction

e standard deviation σ of an array of m data Xi is the simplest measure of how spread thesenumbers are around their average value X = (1/m)

∑mi=1Xi. Suppose that the Xi’s represent

nal Physics marks of m students of a high-school. Most worrisome scenarios for the headmas-ter would be a low X and/or a high σ, signaling an overall poor and/or highly non-uniform

62

63

< L >

O H1 n L

School 1 School 2

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ç

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çç

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ç

ç

ç

ç

ç

ç

ç

ç

ç

ç

ç çç

ç

ç

ç

ç

ç

ç

ç

ç çArts

Physics

á

á

á

á

áá

á

á

á

á

á

á

á

á

á

á

á

áá

áá

á

á

áá

á

á á

á

ááá

á

á

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Arts

Physics

L

Figure 5.1: Top: Sketch of the probability density of the likelihood ratio L of a Gaussian i.i.d. data set. In yellow,the typical region around the mean of order O(1/n). Larger uctuations are referred to as atypical large deviations.Bottom: Sketch of two multivariate data sets with n = 2 and m = 35. Each point represents a student, for twodierent schools, and his/her marks in Arts and Physics. e two datasets have same generalized variance H , butdierent total variance T . e likelihood ratio L of School 1 is compatible with the i.i.d. hypothesis, while the valueof L for School 2 is atypically far from the average 〈L〉.

performance.What if Physics and Arts marks are collected together? Detecting performance issues now

immediately becomes a much harder task, as data may uctuate together and in dierent directions.A two-dimensional scaer plot may help, though. e “center” of the cloud gives a rough indicationof how well the students perform on average in both subjects. But how to tell in which subjectthe gap between excellent and mediocre students is more pronounced, or whether outstandingstudents in one subject also excel in the other?

In Fig. 5.1 (Boom) we sketch two scaer plots of marks (School 1 and School 2) adjusted tohave zero mean. A meaningful spread indicator seems to be the shape of the ellipse enclosing eachcloud. For example, an almost circular cloud - like School 1 - represents a rather uninformativesituation, where your Arts marks tell nothing about your Physics skills, and vice versa. Conversely,a rather elongated shape - like School 2 - highlights strong correlations between each student’smarks in dierent subjects. e semi-axes of the ellipse are called components, and are givenby the spectral decomposition of the covariance matrix of data. Finding the most signicantcomponents for a high-dimensional dataset is the goal of Principal Component Analysis [123, 199],with countless applications in science.

For a bunch of many scaered points it would be desirable to summarize the overall spreadaround the mean just by a single scalar quantity, like the perimeter or area of the enclosing ellipse.Not surprisingly, however, these indicators (taken individually) have evident shortcomings [121].

64

Surely a wiser choice is to combine more than one single measure of spread (like perimeter orarea alone), to obtain a more revealing indicator.

In this Chapter, by using the machinery developed in Chapter 4 on the Coulomb gas method,we will study the joint statistics of “perimeter” and “area” enclosing clouds of random high-dimensional data. Why this is a crucial (and so far unavailable) ingredient for an accurate dataanalysis will become clearer very shortly.

In the more general seing of n subjects and m students (n variates and a sample size m),their marks can be arranged in a n ×m matrix X , adjusted to have zero-mean rows (see theSubsection 1.3.3 on multivariate statistics). We then construct the normalized n× n covariancedata matrix S = (1/n)XX †, with non-negative eigenvalues λ = (λ1, . . . , λn), which is preciselythe multi-dimensional analogue of the variance σ2 for a single array. e surface and volume(“perimeter” and “area” in the two-dimensional example) of the enclosing ellipsoid are related to

T =1

n

∑i∈[n]

λi and G =∏i∈[n]

λ1/ni , (5.1)

the scaled trace and determinant of S . In statistics, these objects are called total and generalizedvariance respectively [231, 9]. As discussed before, blending both estimators together would bepreferable, like in the widely used positive scalar combination

L = T −H − 1 , (5.2)

called likelihood ratio [9], where

H = logG =1

n

∑i∈[n]

log λi . (5.3)

Values of L for dierent shapes of the cloud of data are sketched in Fig. 5.1 (Boom).Now, suppose that we wish to test the hypothesis that the dataXij (yielding a certain empirical

L) are independent and identically distributed. To this aim, we compare our data with a randomn×m dataset Xij independently drawn from a standard Gaussian distribution, our reference(null) model. e correspondingL has a probability density (see Fig. 5.1 Top) peaked on a region of∼ O(1/n) around its mean (see below). Standard statistical tools [231, 9] then allow to assess thelikelihood that L has been drawn from the density of L, thus rejecting or not the null hypothesiswithin a certain condence interval. However, what if an atypically high or low L (with respectto a null i.i.d. model) comes out from the data? We would be tempted to reject the test hypothesisoutright. However, this might lead to a misjudgment, as atypical values of L for the null modelcan (and do) occur (just very rarely). What is the probability of this rare event? e improvedCoulomb gas technique introduced in the previous Chapter, is tailored to give a solution to thisproblem, by computing the joint statistics of total and generalized variance for a large randomdataset. Besides providing an elegant solution to a challenging problem, our unifying frameworkrecovers some partial results earned by other techniques in the statistics literature.

65

5.2 Setting and summary of results

We consider n×mmatricesX whose entries are real, complex or quaternion independent randomvariables with standard Gaussian densities labelled by Dyson’s index β = 1, 2 and 4 respectively,and we form the (real, complex or quaternion) covariance (Wishart) matrix [237]

S =1

nXX † . (5.4)

We dene the rectangularity parameter c = m/n ≥ 1 and we recognize in (5.4) the Wishart-Laguerre ensemble (2.123) of parameter c discussed in the Example 2.3 of Section 2.6. e jointprobability density of the n real and positive O(1) eigenvalues of S is

P(λ) =1

Zne−β[−

∑i<j log |λi−λj |+n

∑k V (λk;α,ξ)] (5.5)

where the energy function E(λ;α, ξ) in the exponent contains the external potential

V (λ;α, ξ) =

−α log λ+ ξλ , forλ > 0 if α > 0 (or λ ≥ 0 if α = 0)

+∞ otherwise(5.6)

with α =c− 1

2+

1− 2/β

2n= sc +O

(1

n

)and ξ = 1/2 . (5.7)

e normalization constant Zn =´

(0,∞)ndλ e−βE(λ;α,ξ) is known for nite n from the celebrated

Selberg integral [91, 8]. e joint law of the eigenvalues (5.5) is the Gibbs-Boltzmann weight of a2D Coulomb gas constrained to the positive half-line and subject to the external potential V atinverse temperature β.

e joint probability law of the rescaled linear statistics H = n−1∑

i∈[n] log λi and T =

n−1∑

i∈[n] λi isPn(h, t) = E [δ (h−H) δ (t− T )] , (5.8)

where the average is with respect to the law (5.5). Its Laplace transform

Pn(s, w) = E[e−βn

2(sH+wT )]

(5.9)

can be easily wrien (for every n) as the ratio of two partition functions

Pn(s, w) =Zn(s, w)

Zn(0, 0), (5.10)

withZn(s, w) =

ˆ(0,∞)n

dλ e−βE(λ;α−s,ξ+w) , Zn(0, 0) ≡ Zn , (5.11)

66

and is nite for s ≤ (c− 1)/2 +O(n−1

)and w > −1/2. From now on we will introduce the

notation sc ≡ (c− 1)/2. Here we show that, for large n and m, with c = m/n ≥ 1 xed, thisLaplace transform behaves as

Pn(s, w) ≈ e−βn2J(s,w) , (5.12)and we compute explicitly the cumulant generating function (GF)

J(s, w) = limn→∞

− 1

βn2Pn(s, w) (5.13)

using a Coulomb gas technique. e main result of this Chapter is the following.

eorem 5.1 (Joint large deviation function of Generalized and Total Variances). Let λ =(λ1, . . . , λn) be distributed according to (5.5)-(5.6) and letH = n−1

∑i∈[n] log λi andT = n−1

∑i∈[n] λi.

eir joint cumulant generating function J(s, w) dened by (5.13) exists for

s ≤ sc ≡c− 1

2and w > −1

2(5.14)

and is given byJ(s, w) = JH(s) + JT (w)− s log (1 + 2w) , (5.15)

where JT (w) and JH(s) are the individual GF of cumulants of T and H separately

JH(s) = φ (s− sc)− φ (−sc) , (5.16)

JT (w) =c

2log (1 + 2w) , (5.17)

with φ(x) = −32x+ x2 log(−2x)− (1−2x)2

4 log (1− 2x) for x ≤ 0.

e proof of this eorem is postponed in Section 5.5. To make the exposition more clear wewill analyze the large deviation function of T and H separately, and then we will turn to theirjoint behavior. Again, for later convenience we will use the notation

H[µ] =

ˆdµ(λ) log λ and T [µ] =

ˆdµ(λ)λ . (5.18)

5.3 Scaled trace

As an instructive example of the Coulomb gas technique, let us consider the detailed steps for thelarge deviation functions JT (w) (cumulant GF) and ΨT (t) (rate function) of the total varianceT = n−1

∑i λi. Indeed, in this case there are no technical obstructions and we easily deduce the

following result.

eorem 5.2 (Large deviation principle for the total variance). Let us consider the linear statisticsT = n−1

∑i λi with λ = (λ1, . . . , λn) distributed according to the probability law (5.5). en, T

satises a large deviation principle with speed βn2 and rate function

ΨT (t) =t− c

2+c

2log(ct

)for t ∈ (0,∞). (5.19)

67

Proof. e external potential V in (5.6) grows suciently fast to guarantee (by eorem 3.1 ofSection 3.1) that the empirical spectral density satises a large deviation principle with speed βn2

I[µ] = EV [µ]− EV [%?] , (5.20)

with

E0[µ] = −1

2

ˆdµ(λ)dµ(λ′) log|λ− λ′|+ 1

2T [µ]− scH[µ] , (5.21)

E0[%?] = infµ∈M([0,+∞))

E0[µ] , (5.22)

where we have consider only the leading terms of the external potential V (λ;α, ξ) (recall thatsc = (c− 1)/2). e map µ ∈ M([0,+∞)) 7→ T [µ] ∈ R is continuous. en, the contractionprinciple predicts a large deviation principle for T = T [µ] with the same speed βn2 and ratefunction

ΨT (t) = infµ∈M([0,+∞))

I[µ] : T [µ] = t . (5.23)

As discussed in the previous Chapter, the shortened version of the Coulomb gas method goesthrough the evaluation of the most probable conguration of the Coulomb gas with a prescribedvalue t of the scaled trace T . is constraint is implemented by introducing a Lagrange multiplierw and minimize the functional

Ew[µ] = E0[µ] + wT [µ] , (5.24)

with E0[µ] in (5.21). e corresponding saddle point equations are

%?w(λ′)

λ− λ′dλ′ = w + 1

2 −scλ ,

ˆdλ %?w(λ)λ = t ,

ˆdλ %?w(λ) = 1 .

(5.25)

We multiply the rst equation by λ and we integrate with respect to %?w(λ)dλ using the identity(2.68) to get

1

2=

(w +

1

2

) ˆdλ %?w(λ)λ− sc

ˆdλ %?w(λ) . (5.26)

Using the normalization condition and the constraint in (5.25) we get the relation between theLagrange multiplier w and t at the saddle point

t?(w) ≡ T [%?w] =c

1 + 2w. (5.27)

68

e positivity condition t ≥ 0 imposes w > −1/2, according to what we expect. e scaledcumulant GF obeys the relation dJT (w) = T [%?w]dw with JT (0) = 0:

JT (w) =

ˆ w

0

c

1 + 2w′dw′ =

c

2log (1 + 2w) . (5.28)

Similarly, one immediately gets the rate function by integrating w(t) (the inverse of (5.27)):

ΨT (t) = −ˆ t

t(0)w(t′)dt′ =

ˆ t

c

1

2

(1− c

t′

)dt′ (5.29)

=t− c

2+c

2log(ct

)for t ∈ (0,∞), (5.30)

proving (5.19).

e probability law of the scaled trace T satises a large deviation principle of the formPT (t) ≈ e−βn2ΨT (t), with ΨT (t) in (5.19) analytic for t ∈ (0,+∞). We identify three regimes:

i) typical uctuations of order O(1/n) are governed by the quadratic behavior of ψT (t)

around t?(0) = c, implying an asymptotically Gaussian law T ∼ AN (c, βn2

2c );

ii) large deviations for t c exhibit a c-independent exponential decay PT (t) ≈ e−βn2t/2;

iii) for t c we nd a c-dependent power law PT (t) ≈ tβn2c/2.

On overall, one can remember the following scheme:

PT (t) ≈ e−βn2ΨT (t) ∼

tβn

2c/2 , (t→ 0) ,

e−βn2 (t−c)2

4c , (t ∼ c) ,e−βn

2t/2 , (t→ +∞) .

(5.31)

ese predictions have been conrmed by extensive numerical investigations. A sample sizeof about N = 108 complex (β = 2) Wishart matrices has been eciently generated using thetridiagonal construction [71]. e data are ploed in Fig. 5.2 and show an excellent agreementwith the large deviation function (5.19) of eorem 5.2.Remark 5.1. A careful computation of a rate function by the Gartner-Ellis theorem should bealways veried a posteriori. Indeed, a priori, the Legendre-Fenchel transform infw[JT (w)− wt]is the convex envelope of the rate function ΨT (t) (that can be dierent from the “true” ΨT (t)).However, we can verify that in our problem JT (w) (which is dened on an proper interval of theline) satises both the dierentiability and steepness conditions (see Section 3.1).

69

Gaussian

Rate Function

çç Simulations

çç

ççç

çççççççççççççççççççççççççççççççççççççççççççççççççççççççç

ççççççç

çççççç

çççç

çççç

çççç

çççç

çççç

çççç

çççç

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ç

-7ΣT -5ΣT -3ΣT -ΣT 0 +ΣT +3ΣT +5ΣT +7ΣT

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

t-E@TD

-HΒ

n2 L-

1lo

gP

THtL

Figure 5.2: Numerical simulations (black circles) for the total variance T = n−1 ∑i λi for n = 15 complex (β = 2)

Wishart matrices. Here the sample size is N = 2.5 · 108. e Gaussian approximation (orange line) with averageE[T ] (5.63) and standard deviation σT =

√Var(T ) (5.64) ts well the data for small uctuations of order ∼ 3σT but

deviates strongly for atypical uctuations. e global behavior is captured by the large deviation function (blue line)ψT (t) of (5.19).

5.4 Scaled log-determinant

For the scaled-determinant, we preliminarly observe that the map µ ∈M([0,+∞)) 7→ H[µ] ∈ Ris continuous on the measures with suppµ ⊂ (0,+∞). In this case, indeed, we will see thatsome problems will arise with measures µ whose support contains the origin 0 ∈ suppµ. Forthis problem we are looking for the equilibrium measure of

Es[µ] = E0[µ] + sH[µ] . (5.32)

e saddle-point density %?w = arg min Es[µ] is the solution of

%?s(λ

′)

λ− λ′dλ′ = 1

2 − (sc − s) 1λ ,

ˆdλ %?s(λ) log λ = h ,

ˆdλ %?s(λ) = 1 .

(5.33)

70

For s ≤ sc, the net external potential V (λ;α, ξ)+s log λ is convex and, using the Tricomi formulaone nds the following single support solution

%?s(λ) =1

2πλ

√(λ− λ−) (λ+ − λ) 1 λ−<λ<λ+ , (5.34)

h?(s) ≡ H[%?s] = ϕ (s− sc) (5.35)

where the edges of the support are

λ±(s) =(

1±√

1− 2 (s− sc))2

, (5.36)

andϕ(x) = −1 + (1− 2x) log (1− 2x) + 2x log (−2x) . (5.37)

e cumulant GF JH(s), is readily obtained by elementary integration

JH(s) =

ˆ s

0ds′H[%?s′ ] (5.38)

= φ(s− sc)− φ(−sc) , (5.39)

with φ(x) = −32x+ x2 log(−2x)− (1−2x)2

4 log (1− 2x) for x ≤ 0. In this case, we are not ableto invert h?(s) in terms of elementary functions and then we write

ΨH(h) = −ˆ h

h?(0)dh′s?(h′) , where h?(0) = −1− (c− 1) log(c− 1) + c log c . (5.40)

Remark 5.2. As for the scaled trace T , here the cumulant GF of the log-determinants is nite on aproper interval of the real line. In this case we have JH(s) nite and continuous with its rstderivative for s ∈ (−∞, sc]. e image of h(s) covers only part of the range of H and in this cas, the scaled cumulant GF is non-steep because its derivative is bounded at the edge of its domainlims↑sc J

′H(s) = −1. In this case we can use only a “local” Gartner-Ellis theorem to conclude that

the rate function is ΨH(h) = infs[JT (s)− sh] in (5.40) for h > −1.However we can again identify three regimes:

i) a Gaussian regime around the mean h?(0);

ii) atypical right deviations of H (h → ∞) are super-exponentially suppressed PH(h) ≈exp(−(βn2/2)eh) (independent of c);

iii) the statistics of anomalously small log-determinants (h h?(0)) cannot be captured bythe cumulant GF which fails to be steep. Small log-determinants are described by a dierentmechanism (see below).

71

For any β > 0 and c > 1 we can summarize the large deviation principle as

PH(h) ≈ e−βn2ΨH(h) ∼

e−βn2 [t−(−1−(c−1) log(c−1)+c)]2

4 log cc−1 , (h ∼ h?(0))

e−βn2 eh

2 , (h→ +∞) .

(5.41)

In particular, the non-Gaussian corrections for H account for deviations of the scaled determinantG =

∏ni=1 λ

1/ni from the log-normal distribution. Indeed, since H = logG, one immediately can

relate the densities of G and H as

PG(g) = PH(log g)/g . (5.42)

However, more directly one observes that the following relation holds PH(s) = MG(1 − s),where PH(s) is the Laplace transform of H andMG(s) is the Mellin transform of G

MG(s) = E[Gβn

2(s−1)]. (5.43)

Moreover, the Mellin transform evaluated at s = 1 + k/βn2 provides the kth moment E[Gk]of the scaled determinant. en, as a bonus, it is possible to deduce the moments of the scaleddeterminant:

E[Gk] ≈ exp[−βn2JH

(−k/βn2

)](5.44)

= e−kE[H]

[1 +

k2

βnlog

(c

c− 1

)+ o

(n−1

)], (5.45)

with E[H] = −1− (c−1) log(c−1)+c log c, as in (5.40). eO(1) term was found in [49] usingan asymptotic analysis of the exact Mellin transform. For statistics of determinants of randommatrices, see also [49, 215, 138].

5.4.1 Smaller H : a dierent mechanism

We have seen that for the log-determinant the (local) Gartner-Ellis theorem yields only part ofthe rate function ΨH because the image of h(s) for s ≤ sc does not cover the whole range of H .As s reaches the critical value s = sc, the lower edge λ− of the saddle-point density (5.58) hitsthe origin, and %?s acquires an inverse square-root singularity there

%?sc(λ) =1

2π

√4− λλ

1λ∈(0,4) . (5.46)

For s > sc the Laplace transform is no longer nite and this critical point corresponds to ac-independent critical value for the log-determinant Hcrit = −1 (or Gcrit = 1/e).

If we pretend to somehow extend the analysis in the Laplace domain for s > sc, we havethe following scenario. Beyond s = sc, the logarithmic part of the external potential V (λ) =12λ + (s − sc) log λ becomes aractive. With the convention that Coulomb gas is negatively

72

charged, this scenario corresponds to have an eective positive charge is located at the originλ = 0. In this conditions, the only equilibrium conguration of the gas is the following: a nitefraction of the Coulomb gas (the closest to the positive charge at the origin) condenses at the originneutralizing the external positive charge and leaving behind a bulk of “macroscopic” eigenvaluesstill ∼ O(1).

Seing c = 1 for simplicity, motivated by the previous heuristic discussion we may look for aminimizer of energy function in the form

%?s(λ) = pδ(λ− λ0) + %s(λ), (5.47)

with p ≡ p(s) ∈ (0, 1) and λ0 ≡ λ0(s) ≥ 0. e saddle-point equations should read

dλ′

pδ(λ′ − λ0) + %s(λ′)

λ− λ′=

1

2+s

λ,

ˆdλ [pδ(λ− λ0) + %s(λ)] log λ = h ,

ˆdλ[pδ(λ′ − λ0) + %s(λ

′)]

= 1 .

(5.48)

From the normalization condition we immediately obtain´

dλ %s(λ) = 1 − p. Expanding therst equation we get

p

λ− λ0+

dλ′

%s(λ′)

λ− λ′=

1

2+s

λ(5.49)

and one may verify that this %? is a solution of the saddle-point equation in the limit λ0 → 0 andp→ s with %(λ) = (2π)−1

√(4(1− s)− λ)/λ. Note that p = s is exactly the charge necessary

to neutralize the logarithmic potential source at the origin. Since the total negative charge of ourgas is (−n), this neutralization is no longer possible for s > 1 (or s > sc + 1 in the general caseof c ≥ 1).

However, it is easy to understand that this scenario is not the energetically favored one. Indeed,the caveat of the previous analysis is that the logarithmic repulsion of the eigenvalues makes theenergy contribution of the condensed part pδ(λ) divergent! is is the divergent part that makesthe Laplace transform PH(s) not nite for s > sc. Without loss of generality, let us focus againon the case c = 1. At the equilibrium, the density of the Coulomb gas is (5.46) correspondingto the average value of the log-determinant E[H] = −1. We have seen that larger values of Hcorresponds to congurations of the gas supported on [λ−, λ+] with λ±(s) > 0. In this case theGartner-Ellis theorem provides the law of H for H > E[H] = −1. What is the behavior of theCoulomb gas constrained to have H < E[H]? Since the function log λ is divergent for λ ↓ 0, itis reasonable to expect that small values of H = n−1

∑i∈[n] log λi are driven by the statistical

behavior of the smallest eigenvalue λmin. e idea is to split the contribution of the eigenvaluesin two parts:

H[%] =1

nlog λmin +H[%] , (5.50)

73

where%(λ) =

1

n

∑i:λi 6=λmin

δ(λ− λi) . (5.51)

Let us see explicitly what is the law of the smallest eigenvalue:

Pr(λmin > t) =

ˆ(t,+∞)n

dλP(λ)

=1

Zn

ˆ(t,+∞)n

dλe−β[−∑i<j log |λi−λj |+n

2

∑k λk]

=eβn

2t

Zn

ˆ(0,+∞)n

dze−β[−∑i<j log |zi−zj |+n

2

∑k zk]

= e−βn2t (5.52)

where we have used the substitution zi = λi − t, for all i ∈ [n], and we have neglected thesubleading terms in the external potential (α = O(1/n) for c = 1). en, the smallest eigenvalueshas probability density

Pλmin(t) = βn2e−βn

2t 1t≥0 , (5.53)

corresponding to a typical value E[λmin] = 1βn2 (and variance Var(λmin =)(βn2)−2). e

probability density of H can be wrien as

PH(h) =

ˆ +∞

0dtPr(H ∈ [h, h+ dh] |λmin = t)Pλmin

(t) . (5.54)

At this point we should understand the conditioned probability Pr(H ∈ [h, h+ dh] |λmin = t).Typical uctuations of order O(n−2) on the right of E[λmin] are irrelevant for the statisticalbehavior of H . On the contrary the typical uctuations on the le (λmin < E[λmin]) play asignicant role due to the divergent character of log λmin for λmin ↓ 0. Indeed for 0 ≤ t ≤ E[λmin]we have:

PH (h |λmin = t) = PH(h = H

[1

nδ(t− λmin) + %

] ∣∣λmin = t

)= PH

(h =

1

nlog λmin +H [%)]

∣∣λmin = t

)→ PH

(h =

1

nlog λmin +H

[1

2π

√4− λλ

1λ∈(0,4)

]|λmin = t

)

= PH(h =

1

nlog λmin − 1

∣∣λmin = t

)= δ

(h−

(1

nlog λmin − 1

)). (5.55)

74

Roughly speaking, the typical uctuations of order O(n−2) of the smallest eigenvalues do notchange the limiting macroscopic density of the eigenvalues limn n

−1∑

i∈[n] δ(λ−λi). is meansthat the limiting macroscopic density of the eigenvalue conditioned to have H ≤ Hcrit = −1is given by (5.46), irrespective of the value of H ≤ Hcrit. What changes is the location of thesmallest eigenvalues, and this determines the value of the log-determinant byH = 1

n log λmin−1.Using the above result, from (5.54), we easily get

PH(h) ∼ enθ(h+1) , for h −1 , (5.56)

where we have incorporated a phenomenological factor θ. is analysis, completes the picturefor the large deviation of the log-determinant.

5.5 Joint cumulant GF

As promised, the Coulomb gas technique can be exploited to describe the joint statistical propertiesof T and H .

Proof of eorem 5.1. e corresponding variational problem reads

Es,w[%?s,w] = inf%Es,w[%]

= inf%

−1

2

ˆdλdλ′ %(λ)%(λ′) log |λ− λ′|+ (s− sc)H[%] +

(w +

1

2

)T [%]

.

(5.57)

Using the Tricomi formula (2.75) we nd

%?s,w(λ) =2w + 1

2πλ

√(λ− λ−) (λ+ − λ) 1λ−<λ<λ+ , (5.58)

where the edges λ± ≡ λ±(s, w) of supp %?s,w are

λ±(s, w) =

(1±

√1− 2 (s− sc)

)2

1 + 2w. (5.59)

e corresponding values of the scaled log-determinant and trace areh?(s, w) ≡ H[%?s,w] = ϕ (s− sc)− log(1 + 2w) ,

t?(s, w) ≡ T [%?s,w] =c− 2s

1 + 2w,

(5.60)

with ϕ(x) in (5.37). It is immediate to see, puing w or s to zero, that we recover the resultsfor the single linear statistics (5.27)-(5.35). e joint cumulant generating function satises therelation

dJ(s, w) = H[%?s,w]ds+ T [%?s,w]dw , J(0, 0) = 0 . (5.61)

75

and we deduce (through a simple integration)

J(s, w) = JH(s) + JT (w)− s log (1 + 2w) , (5.62)

for s ≤ sc and w > −1/2.

We stress the fact that the joint cumulant GF is not the sum of the single generating functions:T and H are not independent. e joint cumulants of T and H are obtained by evaluating thederivatives of J(s, w) at (s, w) = (0, 0). Extracting the rst cumulants, we obtain the formulaevalid for any β > 0 and c > 1

E[T ] = c, E[H] = −1− (c− 1) log(c− 1) + c log c, (5.63)Var(T )

ω(n, 2)= c,

Var(H)

ω(n, 2)= log

c

c− 1,Cov(T,H)

ω(n, 2)= 1, (5.64)

where we set ω(n, `) =(2/βn2

)`−1. For c = 1 and β = 2 we have found instead

Var(H)

ω(n, 2)= 1 + γ + log n, (5.65)

where γ = 0.57722... is the Euler-Mascheroni constant. e derivation of this new result ispostponed in Section 5.7. In Fig. 5.3 we compare this result with numerical simulations oncomplex (β = 2) Wishart matrices. A standard χ2 test for the comparison of numerical data withour curve gave a reliable condence level of 5%. e logarithmic growth of the variance is theprice of the non continuity of H[·] on measures whose support contains 0. e knowledge of thefull joint cumulant GF shows that T and H are asymptotically jointly normal with mean (5.63)and covariance matrix specied by (5.64). e correlation coecient

r(T,H) =Cov(T,H)√

Var(T )Var(H)=

(c log

c

c− 1

)− 12

, (5.66)

independent of β, is positive for all values of c ≥ 1 and goes to 1 when c→ +∞. e decay ofthe higher order cumulants κ`(T ) and κ`(H) (` > 2) is given for large n by the formulae

κ`(T ) = ω(n, `)c (`− 1)! , (5.67)

κ`(H) = ω(n, `)(`− 3)![(1− c)2−` − (−c)2−`

]. (5.68)

5.6 Likelihood ratio

From P(s, w), it is easy to compute the Laplace transform of the likelihood ratio L = T −H − 1for Wishart random matrices,

PL(s) = E[e−βn

2s(T−H−1)]

(5.69)

= eβn2sP(−s, s) , (5.70)

76

á

á

á

á

á

á

á1 + Γ + log n

áá Simulations

20 21 22 23 24 25 26 27 28 291

2

3

4

5

6

7

8

n

HΒn2 LV

arHHL

Figure 5.3: Rescaled variance of the log-determinant H = n−1 ∑i log(λi) for c = 1. We compare the numerical

data with a logarithmic curve. Each point comes from a sampling of N = 107 complex (β = 2) Wishart matrices. eerror for each point is of order O(10−2) not visible in the picture.

implyingPL(s) ≈ e−βn2J(−s,s)−s , for − 1/2 < s ≤ sc . (5.71)

e cumulants of L follow by simple dierentiations

κ`(L) = κ`(T ) + (−1)`κ`(H) + δκ` with δκ` = ω(n, `)` !

(1− `)θ (`− 1)− δ`,1 , (5.72)

corresponding to a Gaussian behavior on a region around the mean

E[L] = c+ (c− 1) log (c− 1)− c log c , (5.73)

with varianceVar(L) = ω(n, 2) [c+ log (c/(c− 1))− 2] . (5.74)

Note that, since T andH are not independent, the cumulants of L involve the extra term δκ`. Ouranalysis reproduces some known results for T,H and L separately, for β = 1 [122, 12, 124, 118].

5.7 Finite n,m results

e joint Laplace transform Pn(s, w) can be also evaluated exactly at nite n,m, using theLaguerre-Selberg integral [91, 8].

77

á

á

á

á

á

á

á

á

á

á á

ç

ç

ç

ç

çç

ç

çç

çç ç ç ç

JH HsL

JT Hw L

ó

óó ó ó ó ó ó ó ó ó ó ó

0

0-1 2 sc

s

JLH sL

0

0-1 2 sc

s , w

Figure 5.4: Cumulant generating functions JT (w) and JH(s) for w > −1/2 and s ≤ sc from (5.28) and (5.39)respectively. e points are obtained from the nite n formula −(1/βn2) log P(s, w) [?], for n = 8. In the ploedregion the maximal deviation between the exact and the asymptotic expression is of order O(10−4). Inset: CumulantGF JL(s) from (5.71), −1/2 < s ≤ sc of L (again the points are from the nite n result). Here c = 5.

Lemma 5.3 (Selberg’s integral formula, [8]). For all positive numbers a and c

1

n!

ˆ +∞

0· · ·

ˆ +∞

0|∆(x)|2c

n∏i=1

xa−1i e−xidxi =

n−1∏j=0

Γ(a+ jc)Γ ((j + 1)c)

Γ(c). (5.75)

Using (5.75), the Laplace transform

Pn(s, w) = E[e−βn

2(sH+wT )]

(5.76)

=

´(0,+∞)n |∆(λ)|β

∏ni=1 λ

βn(c−12

+1−2/β

2n−s)

i e−βnλi(w+ 12)dλi

´(0,+∞)n |∆(λ)|β

∏ni=1 λ

βn(c−12

+1−2/β

2n

)i e−βnλi/2dλi

, (5.77)

can be easily evaluated as

Pn(s, w) =(βn/2)βsn

2

(2w + 1)δ

n−1∏j=0

Γ[β2 (j + 2n(α− s)) + 1

]Γ[β2 (j + 2nα) + 1

] , (5.78)

78

where δ = βn[n−1

2 + n(α− s) + 1]

and α as in (5.7). e special case w = 0 and α = 0 (squarecase n = m for the log-determinant alone) of (5.78) was considered in [49]. In Fig. 5.4, we showthat our large n formulae reproduce very well the nite n,m result even for moderate values ofn. From (5.78) it is possible to extract the large deviation functions for the scaled trace T (thiscorresponds to puing s = 0), but the asymptotic in the s variable is not trivial.

However we can deduce Var(H) in (5.65). Seing c = 1 (α = 0), β = 2 and w = 0 the (niten) Laplace transform of H can be expressed in terms of Barnes G-functions [16]

PH(s) = E[e−βn

2sH]

= n2sn2n−1∏j=0

Γ [j + 1− 2ns]

Γ [j + 1]

= n2sn2G(2− 2ns)n−2G(1 + n− 2ns)Γ(1− 2ns)Γ(2− 2ns)n−1

G(n+ 1)G(3− 2ns)n−1.

(5.79)

e variance is related to the rst two derivatives of the Laplace transform and these derivationscan be computed explicitly

Var(H) = E[H2]−E[H]2 =1

(2n2)2

P ′′H(0)− P ′H(0)2

=

1

n2[ψ0(n+ 1) + nψ1(n+ 1) + γ] , (5.80)

where ψm(z) = ∂(m+1)z log Γ(z) is the m-Polygamma function. is route provides a nite-n

formula for the variance of H , whose asymptotic is exactly (5.65).It is also instructive to look at the n-Fekete points set, namely the equilibrium conguration

of the Coulomb gas at nite n. In this case the equilibrium conditions of the gas with the twoconstraints

∑i λi = nt and

∑i log λi = nh read

∑j(6=i)

1

λi − λj+α− sx−(w +

1

2

)= 0 , ∀ i ∈ [n] ,∑

i

λi = nt ,∑i

log λi = nh .

(5.81)

We have already seen in the Example 2.3 in Section 2.6 that the solution λ? = (λ?1, . . . , λ?n) is

provided by the zeros of an associated Laguerre polynomial. Indeed, the polynomial P (x) =∏i(x− λ?i ) solves the dierential equation

xP ′′(x)− n [(1 + 2w)x− (α− s)]P ′(x) + (1 + 2w)n2P (x) = 0 . (5.82)

Any polynomial solution of (5.82) is a multiple of the associated Laguerre polynomialLn(α+s)−1n (nx(1 + 2w)).

e equilibrium positions of the Coulomb gas are the solutions of Ln(α+s)−1n (nx(1 + 2w)) = 0.

79

Recalling the denition of the Laguerre polynomials (8.36) we have

Ln(α+s)−1n (nx(1 + 2w)) =

n∑k=0

(n(α+ s+ 1)− 1

n− k

)(−nx(1 + 2w))k

k!=

n∑k=0

akxk , (5.83)

whose zeros satisfy the relation∑(zeroes of Ln(α+s)−1

n (nx(1 + 2w)))

=∑i

λ?i =ak−1

ak, (5.84)

log∏(

zeroes of Ln(α+s)−1n (nx(1 + 2w))

)=∑i

logi λ?i =

a0

ak. (5.85)

An explicit computation provides

1

n

∑i

λ?i →c− 2s

1 + 2w, (5.86)

1

n

∑i

log λ?i → ϕ (s− (c− 1)/2)− log(1 + 2w) , (5.87)

with ϕ(x) in (5.37).

5.8 e classical setting in statistics

It is interesting to investigate the regime c 1, corresponding to n = o(m). is is the classicalseing considered in the statistics literature where for any variate (n) we have a lot of data (m).In our example, we are looking at a scenario where the number of subjects n (Physics, Arts,. . . ) ismuch smaller than the number of students m. We have the following corollary of eorem 5.1.

Corollary 5.4. e linear statistics T andH on Wishart matrices are jointly asymptotically Gaus-sian with mean and covariance structure given by(

TH

)∼ AN

((c

log c

),

2

βn2

(c 11 1

c

)). (5.88)

Proof. Extracting the asymptotics (here the parameter going o to innity is c) of the rst momentsof T,H we obtain √

βn2

2c(T − c) ∼ N (0, 1) (5.89)√

βcn2

2(H − log c) ∼ N (0, 1) , (5.90)

80

that can be rephrased as

T is AN (c,2c

βn2) (5.91)

H is AN (log c,2

βcn2) . (5.92)

ese results (for β = 1) coincide with the results obtained in mathematical statistics with othermethods (see e.g. [162]). However, our analysis also predicts that the quantities (5.89) and (5.90)are jointly Gaussian with covariance

Cov

(√βn2

2c(T − c) ,

√βcn2

2(H − log c)

)= 1 , (5.93)

a result new, to our knowledge. Equations (5.89), (5.90) and (5.89) and (5.93) prove the claim(5.88).

5.9 Comparison between the log-gas and the noninteracting gas

We conclude this Chapter by making a digression to understand the role of the logarithmicinteraction in the Coulomb gas. So far we have considered Coulomb gases at inverse temperatureβ > 0 with Gibbs-Boltmann measure

P int(λ) ∝ e−β[−∑i<j log |λi−λj |+n

∑k

(λi2−(sc) log λi

)], (5.94)

(we are neglecting the subleading terms in n). e superscript denotes that we are referringto an interacting gas. For this log-gas the joint GF of the sum functions T = 1

n

∑i λi and

H = 1n

∑i log λi behave as

P int(s, w) ≈ e−βn2J int(s,w) . (5.95)

We found that J int(s, w) is remarkably independent of the temperature. Now we would like toconsider the noninteracting version of (5.94). We turn o the logarithmic interaction and wedrop o the n scaling in front of the external potential (in order to keep the particle of the gasλi = O(1)). e noninteracting gas at the same temperature has the following Boltzmann-Gibbsweight

P free(λ) ∝ e−β[∑

kλi2−(sc) log λi

]. (5.96)

From the classical theorems of large deviation theory for i.i.d. variables, we know that T and Hsatisfy a large deviation principle with speed βn (to be compared with the βn2 of the log-gas).We would like to make explicit the large deviation functions of the free gas. In this case, by the

81

i.i.d. hypothesis we have

P free(s, w) = E[e−βn(sT+wH)] (5.97)

= E[e−β∑i≤n(sλi+w log λi)] (5.98)

= E[∏i∈[n]

e−β(sλi+w log λi)] (5.99)

=∏i∈[n]

E[e−β(sλi+w log λi)] (5.100)

= E[e−β(sλ+w log λ)]n . (5.101)

e cumulant generating function is readily obtained as

J free(s, w) = − 1

βnlog P free(s, w) = − 1

βlogE

[eβ(sλ+w log λ)

]. (5.102)

Using the denition of the Gamma function

Γ(ν) =

ˆ +∞

0dy yν−1e−y , (5.103)

we easily obtain with elementary changes of variables

E[eβ(sλ+w log λ)

]=

´ +∞0 dλλβ(sc−s)e−λ(

12

+w)´ +∞

0 dλλβsce−λ/2(5.104)

=(β/2)βs

(1 + 2w)β(sc−s)+1

Γ [β (sc − s) + 1]

Γ [βsc + 1]. (5.105)

Let us focus for concreteness on T individually. Puing s = 0 in (5.105) we get

J freeT (w) =

(1

β+ sc

)log (1 + 2w) , (5.106)

and then we get a rate function

ΨfreeT (y) =

y − (c− 1)

2− 1

β+

(c− 1

2+

1

β

)log

(c− 1

y+

2

βy

). (5.107)

Comparing the expression of the large deviation functions (5.106) and (5.107) with their coun-terparts in the log-gas (5.28) and (5.30), we immediately note that for the noninteracting gasboth J free

T (w) and ΨfreeT (y) depend on the temperature 1/β. We have already seen that for the

Coulomb gas the temperature of the gas appears trivially in the large deviation principle as amultiplicative factor in the speed. In the noninteracting picture the role of β is more delicate. eobservable consequences are in terms of the cumulants. As an example, while in the log-gas the

82

average is independent on the temperature and the variance scales like 1/β (see (5.63) and (5.64)),here we have

Efree[T ] = c− 1 +2

β, Varfree(T ) =

2

βn

(c− 1 +

2

β

). (5.108)

For instance, for c = 1 (this corresponds to have i.i.d. exponential random variables λi ∼Exp(β/2) in the uninteracting gas) we have

Eint[T ] = 1 , Varint(T ) =2

βn2, (5.109)

Efree[T ] =2

β, Varfree(T ) =

4

β2n. (5.110)

Chapter 6

Joint statistics of quantum transportin chaotic cavities

6.1 Introduction

We consider the problem of quantum (electronic) transport in mesoscopic devices. e typicalseing is a cavity etched in semiconductors and connected to the external world by two aachedleads. e cavity is brought out of equilibrium by an applied external voltage.

Random matrix theory has provided tools and invaluable insights to successfully describe theuniversal behaviour of these systems [23]. In the RMT approach, the scaering process inside thecavity is governed by a scaering S-matrix drawn at random from the unitary group [15, 116](see also [100, 133, 170] for most recent analytical results on distribution of S).

e two incoming leads may in general support n1 and n2 (with n1 ≤ n2) open electronicchannels, i.e. dierent wave numbers of the incoming electronic modes. e S-matrix, having anatural block form

S =

(r t′

t r′

), (6.1)

in terms of reection (r, r′) and transmission (t, t′) sub-blocks, connects the incoming andoutgoing electronic wavefunctions. Conservation of electronic current implies that S is unitary.Landauer-Buiker theory [136, 86, 115, 39, 40] expresses most physical observables in terms ofthe eigenvalues (λ1, . . . , λn1) of the hermitian matrix tt†. e unitarity condition of S impliesthat 0 ≤ λi ≤ 1, ∀ i ∈ [n1]. Hereaer, we will consider symmetric cavities n ≡ n1 = n2.

Important observables are the dimensionless conductance (measured in units of the conduc-tance quantum G0 = 2e2/h) and shot noise (in units P0 = 2e|∆V |G0 with ∆V the appliedvoltage). ey are expressed as linear statistics on the transmission eigenvalues:

Gn =∑i∈[n]

λi, Pn =∑i∈[n]

λ(1− λi) . (6.2)

Since the eigenvalues are positive and bounded (0 ≤ λi ≤ 1), both Gn and Pn are of order

83

84

O(n). It is then convenient to work with the rescaled conductance G := Gn/n and shot noiseP := Pn/n. More precisely, 0 ≤ G ≤ 1 and 0 ≤ P ≤ 1/4. Since

∑i λ

2i ≤ (

∑i λi)

2 ≤ n∑

i λ2i ,

we have the useful geometric inequalities

0 ≤ P ≤ G(1−G) . (6.3)

e upper bound is aained at G = 1/2, while the lower bound at G = 0 or G = 1.Assuming now that S is a random unitary matrix implies that the λj ’s are random variables.

What is their distribution? When the leads aached to the cavity are ideal (no tunnel barriers),the appropriate choice is to assume S uniformly distributed in one of Dyson’s circular ensemblesof random matrices, labeled by a parameter β: S is unitary and symmetric for β = 1 (for systemsthat are invariant under time-reversal), just unitary for β = 2 (broken time-reversal invariance)and unitary self-dual for β = 4 (in case of anti-unitary time-reversal invariance). In this ideal casethe joint probability density function of transmission eigenvalues λi ∈ [0, 1] (for n ≡ n1 = n2) isgiven [117, 90] by a Jacobi ensemble, namely

P(λ) =1

Zn

∏i<j

|λi − λj |βn∏i=1

λβ/2−1i 10≤λi≤1, (6.4)

where Zn is a normalization constant enforcing´

[0,1]n dλP(λ) = 1. e random nature ofthe transmission eigenvalues λi’s promotes conductance and shot noise to random variablesthemselves, whose statistics is of paramount interest. e average and variance of conductanceand shot noise were considered, using perturbation theory in 1/n, long ago [15, 20, 114, 116]. Inparticular, as n→∞ this variance does not scale with n, as one would naively expects, but insteadaains a constant value depending only on the symmetry class (∝ 1/β), a phenomenon that hasbeen dubbed universal conductance uctuations and that has a clear explaination within the RMTframework. More recently, the classical theory of Selberg integral [91] was employed [192] andaerwards extended [201, 193, 172, 130] to address the analytical calculation of transport statisticsnon-perturbatively (i.e. for a xed and nite number of channels). e full distribution ofG and Pis strongly non-Gaussian for few open channels, with power-law tails at the edge of their supportsand non-analytic points in the interior [157, 130, 131, 201]. For nite n1,2 and β = 2, the Laplacetransform of the full distribution was studied in [178, 179] spoing a connection with integrablesystems and Painleve transcendents. e full distribution (including large deviation tails) for largenwas studied in [222, 223] using a Coulomb gas technique. e statistics of other observables wasstudied in [171, 172, 221, 144, 212, 174]. e integrable theory of quantum transport in the idealcase, pioneered in [178, 179] for β = 2, has been recently completed including the other symmetryclasses [158, 159]. For recent results on the non-ideal case (in presence of tunnel barriers), wherethe distribution of S is instead given by the so-called Poisson kernel [157, 37], see [218, 219, 190].

Here we are concerned with the joint statistics of conductance and shot noise for a largenumber of open channels n and in both regimes of typical and large uctuations. Of particularinterest is the joint law of the scaled conductance and shot noise

P(g, p) = E[δ (g −G) δ (p− P )] , (6.5)

85

(where the average is with respect to (6.4)), and its Laplace transform

P(s, w) = E[e−β2n2(sG+wP )]

=

ˆ[0,1]n

e−β2n(s

∑i λi+w

∑i λi(1−λi))P(λ)dλ. (6.6)

e speed βn2/2 is induced by the speed of the large deviation principle for the empirical spectralmeasure of this random matrix model. In this Chapter we have chosen to factorized β/2 (insteadof β as in the previous Sections) in order to keep in contact with the existing literature.

A nonlinear PDE (connected to the h Painleve transcendent) satised by the logarithm ofP(s, w) in the more general seing of n1,2 nite and generic was discovered for β = 2 in [179],and later generalized to β = 1, 4 [159]. Using this PDE, a nonlinear recurrence for the jointcumulants of conductance and shot noise (still at nite and generic n1,2) was found, togetherwith their leading asymptotic behavior for n1 ' n2 →∞ (technical diculties associated to theextraction of sub-leading terms are discussed in [159]). Earlier, Novaes [172] had computed themoments of conductance alone in terms of a sum over partitions, and Khoruzhenko et al. [130]studied moments and cumulants of conductance and shot noise for β = 1, 2.

e arsenal of sophisticated techniques employed to tackle the calculation of joint cumulantsfor nite n1,2 is truly impressive. In physical terms, however, it would probably be desirable to geta more intuitive understanding, based on simpler and more immediately decipherable formulae.Here we use the physically transparent Coulomb gas method to get a neat picture of the mutualrelations between conductance and shot noise in such systems. e obtained formulae for the ratefunctions are expressed in terms of elementary functions only, and allow to recover eortlesslythe leading terms of the cumulants earned by other methods. e full phase diagram in the (G,P )plane is obtained exactly and linked to the equilibrium electrostatic properties of the associated2D Coulomb gas.

6.2 Summary of results

e large deviation theory of Coulomb gas systems predicts that the scaled conductance andshot-noise G and P satisfy a large deviation principle. More precisely, their joint law P(g, p) andLaplace transform P(s, w) behave asymptotically in n as:

P(g, p) ≈ exp

(−β

2n2Ψ(g, p)

), (6.7)

P(s, w) ≈ exp

(−β

2n2J(s, w)

), (6.8)

and both the joint rate function Ψ(g, p) and the joint cumulant generating function J(s, w) areindependent of β. Here we will outline the key steps to compute them. We nd that J(s, w)(and correspondingly Ψ(g, p)) may take ve dierent functional forms in dierent regions ofthe (s, w) (respectively, (g, p)) plane. is is a direct consequence of phase transitions in the

86

associated Coulomb gas problem (see next Section). Across the lines of phase separation, the setof derivatives of order three of the cumulant GF (the free energy of the associated Coulomb gas)is discontinuous, as it is customary in this type of problems (see [153] for a recent review). erate function Ψ(g, p) has a global minimum (zero) for g = 1/2 and p = 1/8, corresponding tothe average value of conductance and shot noise for large n, E[Gn] = n/2 and E[Pn] = n/8.Expanding the rate function around this minimum, we nd that the typical joint uctuations ofconductance and shot noise are asymptotically Gaussian, with a diagonal covariance matrix (see(6.50)) implying absence of cross-correlations to leading order. is is in agreement with earlierndings [193, 159, 179, 130, 171, 55]. However, atypical uctuations far from the average are notdescribed by the Gaussian approximation, but rather by the full large deviation functions. Forw → 0 (resp. s→ 0) the joint cumulant GF J(s, w) reduces to the individual generating functionof conductance (resp. shot noise), computed in [222, 223]. In the next Section, we set up theCoulomb gas calculation.

6.3 e associated Coulomb gas problem

e measure (6.4) is the Gibbs-Boltzmann measure of a Coulomb gas at inverse temperatureβ/2 > 0

P(λ) =1

Zne−

β2 [−

∑i 6=j log |λi−λj |+n

∑nk=1 V (λk)] (6.9)

in the external potential

V : R −→ (−∞,+∞] : x 7→

0 +O(1/n) if 0 < x < 1

+∞ otherwise. (6.10)

e potential of the Gibbs-Boltzmann weight is an extended valued single particle function,bounded from below and equal to +∞ outside the support (P(λ) has support in Λn withΛ = [0, 1]). In this case, the growth condition (2.55) is satised. en the equilibrium measureof this Coulomb gas exists, is unique and has compact support. We notice that the externalpotential (6.10) in the well [0, 1] is immaterial in the large n limit (it vanishes as n increases). Wealready computed the equilibrium measure µV of this gas in the example 2.4 of Section 2.6 andwe found that it is absolutely continuous dµV (λ) = %?(λ)dλ, the saddle-point density being thearcsine law

%?(λ) =1

π√λ(1− λ)

1 0<λ<1 . (6.11)

87

e contraction principle predicts that the joint rate function Ψ(g, p) of (G,P ) is

Ψ(g, p) = inf%

E0,0[%]− E0,0[%?0,0] : G[%] = g andP [%] = p

(6.12)

= infs,w

inf%

E0,0[%]− E0,0[%?0,0]− s(g −G[%])− w(p− P [%])]

(6.13)

= infs,w

inf%

Es,w[%]− E0,0[%?0,0]− (sg + wp)]

(6.14)

= infs,w

Es,w[%?s,w]− E0,0[%?0,0]− (sg + wp)]

(6.15)

= infs,w

[J(s, w)− (s, w) · (g, p)] , (6.16)

where we have introduced the energy functional

Es,w[%] := −ˆ 1

0

ˆ 1

0dλdλ′ %(λ)%(λ′) log |λ− λ′|+

ˆ 1

0dλ %(λ) [sλ+ wλ(1− λ)] , (6.17)

and, as usual, we use the notation

G[%] :=

ˆ 1

0dλ %(λ)λ , P [%] :=

ˆ 1

0dλ %(λ)λ(1− λ) . (6.18)

Here we are tacitly assuming that the minimization problem can be restricted to the set ofabsolutely continuous measures without penalization:

inf%Es,w[%] = Es,w[%?s,w] , (6.19)

(where %?0,0 is the above mentioned arcsine law). is is justied a posteriori by the uniqueness ofthe equilibrium measure. We stress the fact that both G[·] and P [·] are continuous maps on thecontinuous measures inM([0, 1]) and s, w can range over all the real axis (the Laplace transformPn(s, w) is nite for all s, w ∈ R). en we are safe to apply the contraction principle. In Laplacespace the associated minimization problem is

J(s) = inf%

Es,w[%]− E0,0[%?0,0]

(6.20)

Hereaer, for every s and w we dene

V (λ; s, w) =

sλ+ wλ(1− λ) if 0 < λ < 1

+∞ otherwise(6.21)

Suppose that the equilibrium measure of Es,w is absolutely continuous %?s,w(λ)dλ. en, fromeorem 2.5 there exist a constant C ∈ R such that

(i) V (λ; s, w)−´

dλ′ %?s,w(λ′) log |λ− λ′| ≥ C , ∀x ∈ R;

(ii) V (λ; s, w)−´

dλ′ %?s,w(λ′) log |λ− λ′| = C , onx : %?s,w(x) > 0

.

88

Aer a standard weak dierentiation with respect to λ, we obtain the singular integral equationfor %?s,w

d

dλV (λ; s, w) =

%?s,w(λ′)

λ− λ′dλ′, (6.22)

for almost every λ where a charge exists %?(λ) > 0. Once again, in the framework of ourelectrostatic model, (6.22) is the continuous version of the force balance condition for the chargecloud %?s,w(λ) to be in equilibrium.

First, it is worth noticing a symmetry of this problem. e external potential is invariant (upto an immaterial constant) under the exchange (s, λ)→ (−s, 1− λ), namely,

V (1− λ;−s, w) = V (λ; s, w)− s. (6.23)

Since the logarithmic repulsion is also invariant under the transformation λ→ 1− λ, the energyfunctional Es,w inherits this symmetry. erefore, the equilibrium spectral densities correspondingto (s, w) and (−s, w) are simply related by %?−s,w(λ) = %?s,w(1 − λ). us the phase diagram(s, w) is invariant under the inversion s→ −s and it is sucient to study the problem for s ≥ 0.In next Section, we compute the explicit solutions of the singular integral equation (7.72) for anygiven value of (s, w) and we nd that ve dierent functional forms (phases) are possible for %?s,win various regions of the (s, w) plane.

VHΛ; s,wL-LogÈΛ-Λ'È

æ æ æ æ æ ææ æ æ æç ç ç ç ç çç ç ç ç

0 1Figure 6.1: 2D Coulomb gas system associated to the transmission eigenvalues λi’s with constrained G and P . eCoulomb gas is conned between 0 and 1 and is in equilibrium under the logaritmhic interaction and the externalpotential (arbitrary units). e points come from a Montecarlo simulation. Here n = 10, s = −5 and w = −25.

89

6.4 Solution of the variational problem

e saddle-point equation (6.22)

s+ w − 2wλ = 2

supp %?s,w

%?s,w(λ′)

λ− λ′dλ′ (6.24)

can be easily solved when the potential V (λ; s, w) = sλ+wλ(1−λ) is convex in [0, 1], by meansof the Tricomi formula. In this case, the saddle-point density has connected support (Lemma 2.6)supp %?s,w = [a, b] and the Tricomi formula (2.77) predicts that

%?s,w(λ) =1

π√

(b− λ)(λ− a)

[1− 1

π

b

a

√(b− λ′)(λ′ − a)

λ− λ′(s+ w − 2wλ′

)dλ′

], (6.25)

=1 + wλ2 +

(w a+b

2 −s+w

2

)x+

((s+w)(a+b)

4 + w (b−a)2

8

)π√

(b− λ)(λ− a), (6.26)

with 0 ≤ a < b ≤ 1. In this solution we have two arbitrary parameters, the edges of the supporta and b, that should be xed. One can argue that the set of admissible constraints are

a = 0 or %?s,w(a) = 0 (with a > 0) , (6.27)b = 1 or %?s,w(b) = 1 (with b < 1) . (6.28)

We have four combinations of these conditions and therefore four dierent physiognomies of theequilibrium density. We remark that the constraints give the edges of the support as functionsa = a(s, w), b = b(s, w) of s and w.

When the potential V (λ; s, w) is not convex (this happens when w is positive and sucientlylarge compared to s) the saddle-point density could have a disconnected support. However,since V (λ; s, w) has at most two minima for λ ∈ [0, 1], supp %?s,w has at most two connectedcomponents. A common procedure is the following: i) look at the single support saddle-pointdensity using for instance the Tricomi formula; ii) if for some value of (s, w) such a single supportsolution does not exists search for a double support saddle-point density; iii) if there are somevalues (s, w) such that both scenarios are possible (single and double support solution of (6.22))choose the one with lower energy Es,w.

90

6.5 Phase diagram

We have identied ve domains Ωi, 1 ≤ i ≤ 5 in the (s, w)-plane that correspond to ve dierentphases of the Coulomb gas. Depending on the values of (s, w) the support of %?s,w is

supp %?s,w = [0, a] , if (s, w) ∈ Ω1 (6.29)supp %?s,w = [0, 1] , if (s, w) ∈ Ω2 (6.30)supp %?s,w = [a, b] , if (s, w) ∈ Ω3 (6.31)supp %?s,w = [0, a] ∪ [b, 1] , if (s, w) ∈ Ω4 (6.32)supp %?s,w = [b, 1] , if (s, w) ∈ Ω5 (6.33)

with 0 < a < b < 1. e ve dierent domains are shown in Fig. 6.2 and the edge points a, b aredetermined explicitly in each case (in terms of s, w) below.

e physical picture is quite intuitive in terms of the interplay between the logarithmicinteraction between the charges and the potential V (λ; s, w). For moderate values of s and w(i.e. (s, w) ∈ Ω2) the potential is too weak compared to the logarithmic repulsion between thecharges, so that the gas invades the whole interval [0, 1]. For suciently large positive values ofs (Ω1), the potential is strong enough to aract the charges towards λ = 0 and conne the gas in[0, b] ⊂ [0, 1]. In a similar way, large negative values of s (Ω5) provide a repulsive potential thatpushes the gas away from the origin towards the hard wall located at λ = 1. When w < 0 and|s| < −w, V (λ; s, w) has a minimum that aracts the charges; if this well is suciently deep (i.e.(s, w) ∈ Ω3) the Coulomb gas is trapped in the minimum of the potential and does not feel thehard walls. On the other hand, when w > 0 and |s| < −w , V has a maximum that repels thecharges; in Ω4 the peak is suciently high to cause the Coulomb gas to split into two separatedcomponents that tend to stay far apart.

e explicit form of %?s,win these phases is

%?s,w(λ) =q1(λ)

2π

√b− λλ

, if (s, w) ∈ Ω1 , (6.34)

%?s,w(λ) =q2(λ)

2π

1√λ(1− λ)

, if (s, w) ∈ Ω2 , (6.35)

%?s,w(λ) =q3(λ)

2π

√(λ− a)(b− λ) , if (s, w) ∈ Ω3 , (6.36)

%?s,w(λ) =|q4(λ)|

2π

√(λ− b)(a− λ)

λ(1− λ), if (s, w) ∈ Ω4 , (6.37)

%?s,w(λ) =q5(λ)

2π

√λ− a1− λ

, if (s, w) ∈ Ω5 . (6.38)

where qi(λ) are polynomials in λ. In the above expressions a, b and the qi’s depend on s and w,and are given explicitly in (6.61)-(6.62) of the Appendix 6.8. e above formulas are valid for λwithin the support of %?s,w.

91

e constraints that identify a double support solution are oen cumbersome to evaluate.However, if one is only interested in nding the critical line of phase transition between thesingle support phases and the double support phase in the (s, w)-plane, a practical strategy is thefollowing. e condition of null work for moving a charge from a to b,

´ ba %

?s,w(λ)dλ = 0 should

be satised for any double support solution. In the limit this will be valid also at the “birth” ofthe double-support phase when b = 1. en, the requirement

´ 1a? %

?s,w(λ)dλ = 0 with %?s,w(λ)

supported on [0, a?] provides a threshold a? and therefore an equation connecting s and w at the“birth-of-second-cut”, i.e. precisely the line of phase transition in the (s, w)-plane.

For given values of (s, w), the solution %?s,w provides the typical conguration of eigenvaluesyielding prescribed values forG[%?s,w] and P [%?s,w]. In particular, for (s, w) = (0, 0) (uncostrainedproblem) we obtain from (6.35) and (6.61) the arcsine law (6.11) providing the average valuesE[G] = G[%?0,0] = 1/2 and E[P ] = P [%?0,0] = 1/8. In general, (6.35) in Ω2 describes thetypical uctuations around the average values E[G] and E[P ]. Values of (s, w) ∈ Ω1 correspondto a conguration of the transmission eigenvalues (6.34) yielding smaller values of the scaledconductance g E[G] while large deviations g E[G] correspond to (s, w) ∈ Ω5 and aredescribed by (6.38). Similarly, large values for P are given by a typical conguration of eigenvaluesas in (6.36) in phase Ω3, while smaller values for P in phase Ω4 are ascribed to the double supportsolution (6.37). As the point (s, w) moves in the Laplace plane, the conguration of eigenvalues%?s,w(λ) changes according to (6.34)-(6.38). e transition across the critical line γij (the boundaryof two dierent phases Ωi, Ωj) corresponds to a change in shape of the Coulomb gas density,which signals a corresponding phase transition in the extensive quantities G and P . In the nextSection, we use the explicit functional forms of the density in various regions of the (s, w) planeto compute the cumulant GF and the rate functions for this problem.

6.6 Joint large deviation function

e map (s, w) 7→(G[%?s,w], P [%?s,w]

)provides the components of the dierential form

dJ(s, w) = G[%?s,w]ds+ P [%?s,w]dw , (6.39)

where J(s, w) is the cumulant GF in (6.20). For the sake of convenience we will denote by gi(s, w)and pi(s, w) the branches of the functions G[%?s,w] and P [%?s,w] for (s, w) ∈ Ωi:

G[%?s,w] = gi(s, w) if (s, w) ∈ Ωi

P [%?s,w] = pi(s, w) if (s, w) ∈ Ωi .(6.40)

92

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Figure 6.2: Top Le: Phase diagram in the Laplace (s, w) plane. e ve domains Ωi, i = 1, . . . , 5 correspondsto dierent spectral proles %?s,w(λ) of the transmission eigenvalues. e critical lines that separate the phases arereported in appendix in (6.57)-(6.58). Notice that the phases Ω1 (supp %? = [0, b]) and Ω5 (supp %? = [a, 1]) areseparated by the phases Ω2, Ω3 and Ω4. e diagram is symmetric with respect to s. e linesw = 0 (black dotdashed)and s = 0 (gray dashed) have been studied in [222, 223]. We recover their critical points scr = ±4 andwcr = ±8. TopRight: e corresponding phase diagram in the conductance/shot noise (g, p)-plane. e critical points and the liness = 0, w = 0 are represented in the real space for completeness. Bottom: Dierent spectral proles %?s,w(λ) of theCoulomb gas. e analytical curves are superimposed to points from a Montecarlo simulation (Metropolis algorithm) ofa Coulomb gas of n = 30 particles in the external potential V (λ; s, w).

With this notation, from (6.34)-(6.38) we found explicitly:

g1(s, w) =(s+ w −

√(s+ w)2 − 24w

)2

×s+ w + 2

√(s+ w)2 − 24w

432w2, (6.41)

g2(s, w) =1

2− s

16, (6.42)

g3(s, w) =1

2+

s

2w, (6.43)

g5(s, w) = 1− g1(−s, w). (6.44)

93

and

p1(s, w) =(s+ w −

√(s+ w)2 − 24w

)3

×(s+ w)2 − s− 17w + (s+ w − 3)

√(s+ w)2 − 24w

3456w3, (6.45)

p2(s, w) =1

8− w

128, (6.46)

p3(s, w) =1

4+

1

2w− s2

4w2, (6.47)

p5(s, w) = p1(−s, w). (6.48)

e branches g4 and p4 can also be computed but we do not report here their expressions. ecritical lines w = 0 and s = 0 (yielding the GF of conductance or shot noise alone) were studiedin [222, 223] and one can easily recover their results by taking the limit w → 0 (or s→ 0) in theabove expressions (sometimes one should iterate the l’Hopital-Bernoulli rule). e joint cumulantGF J(s, w) follows by a careful integration of the dierential form (6.39), imposing the conditionJ(0, 0) = 0 and the continuity requirement. By inspection of the above expression we argue thatJ(s, w) is not analytic in the whole (s, w) plane. It is however interesting to note that J(s, w) isanalytic in a neighborhood of the origin (s, w) = (0, 0) allowing to obtain by dierentiation thejoint cumulants of (G,P ) to leading order. Indeed we nd

J(s, w) =s

2− s2

32+w

8− w2

256, if (s, w) ∈ Ω2 . (6.49)

erefore, we conclude thatGn and Pn are uncorrelated (to leading order) Gaussian variables withaverage values E [Gn] = n/2 and E [Pn] = n/8, Var (Gn) = 1/(8β), Var (Pn) = 1/(64β), andCov(Gn, Pn) = 0: (

GnPn

)∼ AN

((n2n8

),

(1

8β 0

0 164β

)). (6.50)

Since the higher derivatives of J(s, w) vanish at (s, w) = (0, 0) we also conclude that for thehigher order cumulants

κ`,m(Gn, Pn) = O(

1

n`+m−2

)for `+m > 2 . (6.51)

For generic s andw, the integration of dJ provides the cumulant GF. For instance, for (s, w) ∈ Ω3

the cumulant GF reads:

J(s, w) =3

4+

(s+ w)2

4w+

1

2log(−w

8

)if (s, w) ∈ Ω3 , (6.52)

and one immediately sees that the transition from Ω2 to Ω3 is not analytic since the third-derivativeof J(s, w) is discontinuous at the point (s, w) = (0,−8). In the regions Ω1 and Ω5 that describe

94

large deviations |G| 〈G〉, to the le and to the right respectively, one gets the cumulant GF bydirect integration:

J(s, w) = −5

4+

ˆ (s,w)

(0,−8)dJ(s′, w′) if (s, w) ∈ Ω1,5 , (6.53)

and similarly for the other regions.Remark 6.1. Whenever the real domain (in this case the (G,P )-plane) is bounded, if the GF isdierentiable then it is also Lipschitz continuous. In our case we have

|J(s, w)− J(s′, w′)| ≤ η‖(s, w)− (s′, w′)‖ , with η = sup(s,w)‖∇J‖ ≤ 1 . (6.54)

6.7 Conclusions

In summary, our analysis gives an overall picture of the joint statistics of conductance G andshot noise P for an ideal chaotic cavity supporting a large number n of electronic channels in thetwo aached leads. We employed the random scaering matrix framework and a Coulomb gastechnique in Laplace space to establish the large deviation formulae (6.20) and (6.7), governingthe behaviour of the joint cumulant GF, J(s, w), and the joint rate function Ψ(g, p) of G andP , respectively. ese quantities have been studied separately in [222, 223] by means of thevery same Coulomb gas technique, yielding in the Laplace (s, w)-plane the lines w = 0 (forG) and s = 0 (for P ) that we precisely recover1. Conversely, we are now able to sketch thefull phase diagram in the (s, w) and (G,P ) planes (see Fig. 6.2). We nd that both J(s, w) andΨ(g, p) acquire ve dierent functional forms in dierent regions of their domain. ese dierentexpressions are a direct consequence of phase transitions in an associated Coulomb gas problem.e derivatives of the Coulomb gas free energy (the cumulant GF) across the lines of phaseseparation are discontinuous, a third order phase transition customary in this type of problems[153]. Here, however, we have used the shortcut discussed in Section 4.2. For the reader’s benet,we present here this shortcut in detail for the conductance G. Taking the limit w → 0 in ourexpressions (6.41), (6.42), and (6.44) we easily obtain

g(s) =

1/s s > 4

1/2− s/16 |s| ≤ 4

1 + 1/s s < −4 ,

(6.55)

where g(s) in this case is related to the cumulant GF JG(s) ≡ limw→0 J(s, w) by J ′G(s) = g(s).en, a straightforward integration (with the condition JG(0) = 0) gives

JG(s) =

3/2 + log (s/4) s > 4

s/2− s2/32 |s| ≤ 4

3/2 + s+ log (−s/4) s < −4 .

(6.56)

1For the shot noise, the saddle-point density is symmetric under the transformation λ 7→ 1− λ. is symmetrycan be exploited to obtain an exact result also for the double support solution.

95

Recalling the relation between the cumulant GFJG(s) and the rate function ΨG(g) = maxs [−JG(s) + gs]we easily get Ψ′G(g) = −s(g), where s(g) is the inverse of (6.55). A straightforward integrationprovides the results of [222, 223]. e multidimensional version of this shortcut is the key toobtain results for the joint statistics that would be otherwise inaccessible to the standard versionof the method.

A careful analysis shows that the dierent phases of the Coulomb gas in the (s, w)-planecorrespond to non-analyticities of J(s, w). To be more precise, on the lines γij that separate thephases Ωi and Ωj in the (s, w)-plane, the (at least one) third-derivative of cumulant GF J(s, w)is discontinuous. Our analysis enriches the one of [222, 223] by showing and characterizing thelines of phase transition for the joint probability distribution of the conductance G and the shotnoise P .

6.8 Appendix

Let us denote by γij the critical curve that separates the phases Ωi and Ωj in the (s, w) plane. Wereport their explicit expression

γ13 ∪ γ35 : |s| = −w −√−8w for w ≤ −8 ,

γ12 ∪ γ25 : |s| = w

2+ 4 for − 8 ≤ w ≤ 8

3,

γ24 : s2 + 2w2 − 16w = 0 for 8

3≤ w ≤ 8 , (6.57)

while the critical curves γ14 ∪ γ45 are the solution of the equationˆ 1

b1(|s|,w)q1(λ; |s|, w)

√b1(|s|, w)− λ

λdλ = 0, for w ≥ 4 . (6.58)

More explicitly, γ14 ∪ γ45 is given by:

2wφ1 (b1(|s|, w)) =1

3

(2(|s|+ w) +

√(|s|+ w)2 − 24w

)φ0 (b1(|s|, w)) , (6.59)

for w ≥ 4 with

φk(b) = i

ˆ 1

bdxxk

√b− xx

, 0 < b < 1 . (6.60)

e polynomials qi(λ) and the edges ai, bi of supp%?s,w are given by

q1(λ) =1

3

(−6wλ+ 2(s+ w) +

√(s+ w)2 − 24w

),

q2(λ) = 2wλ2 − (2w + s)λ+w

4+s

2+ 2 ,

q3(λ) = −2w ,

q4(λ) = 2(t− wx) ,

q5(λ) = q1(1− λ;−s, w) , (6.61)

96

and

a1 = 0, b1 =s+ w −

√(s+ w)2 − 24w

3w,

a2 = 0, b2 = 1,

a3 =s+ w +

√−8w

2w, b3 = 1− a3(−s, w),

b4 :

ˆ b4

a4

2(t− wx)

2π

√(a4 − x)(x− b4)

x(1− x)= 0

a4 =s+ (2− b4)w − 2t

w,

t =1

3(s+ 2w − b4w)

− 1

6

√s2 + 2(2b4 − 1)sw − w

(24− w − 8b4w + 8b24w

)a5 = 1− b1(−s, w), b5 = 1, (6.62)

respectively.

Chapter 7

Universal covariance formula forlinear statistics on random matrices

Results on sum functionsAn =∑

i∈[n] a(Xi) abound for i.i.d. random variables. For instance, theclassical Central Limit eorem states that if Var(Xi) is nonzero and nite, then Sn =

∑i∈[n]Xi

is asymptotically Gaussian with “mean” nE[Xi] and variance nVar(Xi). is result can beextended without eort to any sum function An of the Xi’s such that Var(a(Xi)) is nite. If theXi’s are O(1), then both the average E[An] and the variance Var(An) typically grow linearlywith n. But what happens if the n variables are instead strongly correlated? A prominent exampleis given by the n real eigenvalues λi of a self-adjoint random matrix. In this case a completelydierent behavior emerges: if a(x) is suciently smooth, while the average is still of order O(n),the variance aains a nite value for n→∞. Moreover, quite generally Var(An) ∝ 1/β, whereβ (the Dyson index) is related to the symmetries of the ensemble and, on the scaleO(1) of typicaluctuations around the average, the distribution of An is asymptotically Gaussian [183, 45, 14,202, 148, 17, 182, 119, 51, 93]. We have already presented some specic examples in the previousSections. For instance, recalling that the conductance in chaotic cavities can be indeed wrien asa linear statistics of a random matrix, the phenomenon of UCF introduced in Subsection 1.3.3 isreadily understood. e issue of uctuations of generic linear statistics has, however, a longerhistory in the physics and mathematics literature [183, 45, 14, 202, 148, 182, 119, 17, 51, 93], dueto its relevance for a variety of applications beyond UCF, ranging from quantum transport inmetallic conductors [23] and entanglement of trapped fermion chains [217] to the statistics ofextrema of disordered landscapes [151] - to mention just a few.

For a smooth a(x), there exist two celebrated formulas in the physics literature by Dyson-Mehta (DM) [70] and Beenakker (B) [20, 21] for Var(An), the laer precisely derived in thecontext of the quantum transport problem introduced earlier in Chapter 6 (see also [22] for ageneralized (B) formula). ey are deemed universal - not dependent on the microscopic detail ofthe random matrix ensemble under consideration - and correctly predict aO(1) value for n→∞and a universal β−1 prefactor.

What happens now if two linear statistics An =∑

i a(λi) and Bn =∑

i b(λi) are simul-

97

98

taneously considered? Here we set for ourselves the task to nd a universal formula for thecovariance Cov(An, Bn) that would reduce to DM or B for A ≡ B. In Chapter 4 we have seenhow to establish a large deviation principle for joint linear statistics and how to compute thecorresponding large deviation functions. In the specic examples in Chapters 5 and 6 we foundfor Cov(An, Bn) both the freezing of uctuations and the β−1 dependence. It felt natural tond sucient conditions for these phenomena in order to have a conceptual understanding ofuctuations in RMT.

In order to encompass as many cases as possible into a unied framework, we will introducea “conformal map” method. Before turning to the derivation of the main result we will oersome consequences and applications of the covariance formula. In particular, we employ ourformula to probe a quite interesting phenomenon of asymptotic decorrelation (Corollary 7.1),namely for some choices of a(x) and b(x) we get Cov(An, Bn) = o(1). Examples are given fori.) conductance and shot noise in ideal chaotic cavities supporting a large number of electronicchannels (Corollary 9.2), and ii.) uctuation relations for traces of powers of random matrices(Corollary 7.3).

7.1 Setting and results

We consider an ensemble of n×n self-adjoint random matrices Mn whose spectrum in containedin an interval Λ, and whose joint law of the n eigenvalues λ = (λ1, . . . , λn) has the form

Pβ(λ) =1

Ze−β[−

∑i<j log |λi−λj |+n

∑i V (λi)] . (7.1)

Again, we resort to the Coulomb gas analogy: the normalization constant Z =´

Λn dλ e−βE(λ) isthe partition function of a 2D Coulomb gas at inverse temperature β > 0 (the Dyson index) in aconning single-particle potential V (λ), bounded from below and nite for λ ∈ Λ. From now onwe will always assume the following conditions:Assumption 7.1. e potential V (λ) is assumed to be such that the density of states dµ(λ) =%(λ)dλ is absolutely continuous and supported on a single interval σ = supp % of the real line(possibly unbounded).

e form of the joint law of the eigenvalues (7.1) includes classical invariant ensembles [156]such as Wigner-Gauss G, Wishart-LaguerreW , Jacobi J and Cauchy C introduced in Section 2.6.In Table 7.1 the corresponding potentials are listed. We stress, however, that the general seingin (7.1) applies equally well e.g. to non-invariant ensembles such as the Dumitriu-Edelman [71]tridiagonal β-ensembles, for non-quantized values of β > 0. We summarize the connectionwith matrix models in Fig. 7.1 (compare with Fig. 1.1). In this Chapter we will denote as “β-ensemble” any matrix model whose joint law of the eigenvalues has the form (7.1) and satisesthe Assumption 7.1.

Consider now two linear statistics An =∑

i a(λi) and Bn =∑

i b(λi). eir covariance isgiven for any nite n by

Cov(An, Bn) = E[(An −E[An]) (Bn −E[Bn])] . (7.2)

99

i.i.d. entries

Unitary invariant

Coulomb gasOne-cut

Gaussian

P(Mn) =∏ij p(Mij)

P(Mn) = P(UMnU−1)

P(Mn) ∝ e−trVn(Mn)

P(λ) ∝ e−βnE(λ)

Figure 7.1: e one-cut Coulomb gas systems can be realized as the eigenvalues of self-adjoint matrix models. eone-cut Assumption 7.1 selects a portion in our fundamental classication of self-adjoint ensemble. Note that theone-cut Coulomb gas systems can be realized from matrices that are neither i.i.d. nor unitarily invariant (as thetridiagonal models in [71]).

For smooth a(x) and b(x) we show that this covariance (7.2) has the universal form

Cov(An, Bn) =1

βπ2

ˆ ∞0

dk ϕ(k) Rea(k) b∗(k) , (7.3)

with an error term of order O(n−1), which will always be neglected henceforth. Here Re standsfor the real part and ∗ for complex conjugation. e universal kernel ϕ(k) is given by

ϕ(k) =

k tanh(πk) if σ 6= Rk if σ = R

(7.4)

and we have introduced a deformed Fourier transform

f(k) =

ˆ +∞

−∞dx eikxf(T (ex)) , (7.5)

where T is a map dened by the edges of the support σ of the density of states %

T (x) =

xλ−+λ+x+1 for σ = [λ−, λ+]

λ− + 1/x for σ = [λ−,∞)

λ+ − x for σ = (−∞, λ+]

log x for σ = R

(7.6)

100

V (x) Λ σ

G x2

4 (−∞,∞) ±2

W x2 − log x

α2 [0,∞)

(1±√

1 + α)2

J log xα12 (1− x)

α22 [0, 1]

(√1+α2±

√(α1+1)(α1+α2+1)

α1+α2+2

)2

C log√

1 + x2 (−∞,∞) (−∞,∞)

Table 7.1: Summary of various classical matrix models with a joint law of the eigenvalues of type (7.1). For G,W,Jwe provide the edges of the limiting support σ = [λ−, λ+] .

e role of T is to map the positive half-line [0,+∞) to the support σ = supp %. Wheneverσ 6= R, the function T (·) is a conformal map dened by the nite edge(s) of σ. Since no suchconformal mapping exists if σ = R, this case (e.g. the Cauchy ensemble C) must be treateddierently. is is the reason of the evident asymmetry of the covariance formula between thetwo cases. We note that for σ = R, (7.5) is the standard Fourier transform. Eq. (7.3) may be usedwhenever the integral converges.

Let us now oer a few remarks:

(i) formula (7.3) is evidently symmetric upon the exchange A ↔ B, as Cov(An, Bn) =Cov(Bn, An);

(ii) the only dependence on the Dyson index β is through the prefactor β−1 as already antici-pated;

(iii) the details of the conning potential V (x) only appear in the formula (7.3) through theedges λ± of the limiting spectral density %(λ), and not through the range of variability ofthe eigenvalues Λ. is is a consequence of universality of the (smoothed) two-point kernel[7] (see also [34, 35, 94]);

(iv) if the equilibrium measure has compact support σ = [λ−, λ+] (this is always the case whenthe external potential V satises the growth condition (2.55)), the covariance admits thefollowing alternative expression in real space

Cov(An, Bn) =1

βπ2PV

¨ λ+

λ−

dλdλ′φ(λ, λ′)a(λ′)

λ′ − λdb(λ)

dλ, (7.7)

where

φ(λ, λ′) =

√(λ+ − λ)(λ− λ−)

(λ+ − λ′)(λ′ − λ−)(7.8)

and PV stands for Cauchy’s principal value. Formula (7.7), which may be more convenientthan (7.3) in certain cases, reduces for a(x) = b(x) to the generalized B formula for thevariance (as given in [22], Eq. (17)). On the other hand, (7.3) recovers for a(x) = b(x) the

101

DM formula [70] (see Eq. (1.1) in [21]) if σ = (−∞,+∞), and the B formula [20] (see (7.18)below) if σ = [0, 1]. Eq. (7.3) and (7.6) constitute then a neat and unied summary of allpossible occurrences, including the case of semi-innite supports (relevant for some cases[38, 230]).

7.2 Implications of the covariance formula

7.2.1 Asymptotic decorrelation

e representation (7.3) in Fourier space makes apparent that the covariance vanishes to leadingorder e.g. if a(k) is purely imaginary and b(k) is real. e following lemma gives a sucientcondition for such a situation. is condition, i.e. the symmetry of the limiting support σ = −σ,is not articial and occurs quite oen.

Corollary 7.1. Let Mn be a one-cut random matrix model. Suppose that limiting spectral densityof the eigenvalues λi is supported on a symmetric interval (not necessarily nite) around the originσ = −σ. If the functions a ∈ C1 and b ∈ C1 are even and odd respectively, then the covariance ofthe linear statistics An =

∑i∈[n] a(λi) and Bn =

∑i∈[n] b(λi) is arbitrarily small for large n. In

particular, the moments trM `n and trMm

n with ` even andm odd, are asymptotically uncorrelated:

Cov(trM `n, trM

mn ) = o(1) . (7.9)

Proof. e proof relies an the well-known fact that the Fourier transform of an even (odd) functionis real (purely imaginary). By inspection one can verify that the condition

T (1/x) = −T (x) for all x ∈ R (7.10)

is satised if and only if σ = [λ−, λ+] with λ− = −λ+ or if σ = R. In this case (σ = −σ), if thelinear statistics An is dened by an even function a(−x) = a(x), its deformed Fourier transforma(k) is real

a∗(k) =

ˆ +∞

−∞dx e−ikxa (T (ex)) (7.11)

=

ˆ +∞

−∞dx eikxa

(T (e−x)

)(7.12)

=

ˆ +∞

−∞dx eikxa (−T (ex)) (7.13)

=

ˆ +∞

−∞dx eikxa (T (ex)) (7.14)

= a(k) , (7.15)

while if b(−x) = −b(y), then b(k) is purely imaginary. In this case, the covariance formula(7.3) immediately predict that the leading order of Cov(An, Bn) is zero. e statement on themoments is a straightforward consequence.

102

A paradigmatic example of one-cut matrix model Mn with symmetric limiting support σare the ensembles with even potential V (x) = V (−x) like the Gauss-Wigner ensemble G withV (x) = x2/4. We stress the fact that for ensembles with even potential the decorrelation betwennlinear statistics with dierent parity is exact (it is not an asymptotic result but it holds at any niteN ). Corollary 7.1 is an asymptotic result valid for any ensemble with σ = −σ, a more generalcondition. In Fig. 7.2 we oer a numerical demonstration of this decorrelation phenomenon.

12345678

1 2 3 4 5 6 7 8

m

1

HΛ+L+mCov Htr G , tr G mL

0.3

0.2

0.09

0.003

Figure 7.2: Numerical simulations of the rescaled covariance of traces of powers of the Gauss-Wigner ensembleCov(trG`, trGm)/λ`+m+ (for the Gauss-Wigner ensemble λ+ = 2) for some values of ` and m. Here we sampledN = 104 GUE (β = 2) random matrices of size n = 250. e lighter the color the smallest the covariance (in asuitable scale). As predicted by Corollary 7.1, the covariance of traces with dierent parity are zero to leading order(lighter squares in the matrix plot).

7.2.2 Covariance formula in random matrix theory of quantum transport

We focus on the random matrix theory of quantum transport as discussed in the introduction.At low temperature and voltage, the electronic transport in mesoscopic cavities whose classicaldynamics is chaotic can be modeled by a scaering matrix S of the system uniformly distributedon the unitary group [116, 15] (for a review see [23] and references therein). e L× L matrixS is just unitary if time-reversal symmetry is broken (β = 2), or unitary and symmetric incase of preserved time-reversal symmetry (β = 1), where unitarity is required by probability

103

conservation. e size L = n1 +n2 is determined by the number n1,2 of open channels in the twoleads aached to the cavity, and we denote n = min(n1, n2). Many experimentally accessiblequantities can be expressed as linear statistics of the form An =

∑i∈[n] a(λi), where λi ∈ [0, 1]

are the so-called transmission eigenvalues. ey are the eigenvalues of the self-adjoint matrixT = tt†, where t is a min(n1,2)×max(n1,2) submatrix of S and a(x) is an appropriate function.For instance, as already showed in Chapter 6, the dimensionless conductance Gn and shot noisePn of the cavity correspond to the choices a(x) = x and a(x) = x(1 − x) respectively in theLandauer-Buiker theory [136, 86, 41].

e parameter α = n1/n2 − 1 ≥ 0, kept xed in the large n1,2 limit, accounts for theasymmetry in the number of open electronic channels. For a symmetric cavityα = 0. Furthermore,it is well-known that in this seing the transmission eigenvalues λi are distributed accordingto a Jacobi (J ) ensemble [117, 90] with VJ (x) = (α/2) log x, implying an average density %(λ)(see (2.142) with α ≡ α1 and α2 = 0)

%J (λ) =α+ 2

2πλ

√λ− λ−1− λ

1λ∈]λ−,λ+[ , (7.16)

supported on[λ−, λ+] = [α2/(α+ 2)2, 1] (7.17)

(see with Table 7.1).It was precisely in this quantum transport seing that Beenakker’s formula (B) was rst

derived [20, 21]. It reads

Var(An) =1

βπ2

ˆ ∞0

dk|F (k)|2k tanh(πk) , (7.18)

where F (k) =´∞−∞ dx eikx a

(1

1+ex

). It is immediate to verify that (7.18) is recovered from

our (7.3) upon seing a(x) = b(x) and (crucially) α = 0, implying [λ−, λ+] = [0, 1]. If α 6= 0(asymmetric cavities), (7.18) is not applicable and the variance of conductance and shot noisedo depend explicitly on α, in agreement with [192, 193]. From (7.3) one gets the covariances ofconductance Gn and shot-noise Pn to leading order in the channel numbers.

Corollary 7.2. For the matrix model law(Tn) = const× e−βntrV (Tn) with β > 0 and

V : R −→ (−∞,+∞] : x 7→

α2 log x if 0 < x ≤ 1

+∞ otherwise. (7.19)

e conductanceGn = tr Tn and shot-noisePn = tr Tn(1−Tn) are jointly asymptotically Gaussian

104

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áá á á á

á á á á á

Β = 2

Β = 1

0 5 10 15 20

-0.015

-0.010

-0.005

0.000

Α

Cov

HG,P

L

Figure 7.3: Covariance of the dimensionless conductance G and shot-noise P (7.24) as a function of α. e dashedblue (β = 1) and the solid black (β = 2) lines are the analytical results (7.24) found using (7.3). e points are obtainedfrom a numerical diagonalization of n = 104 random Jacobi matrices of size n = 30. For large n the covariance is zerofor symmetric cavities (α = 0) and vanishes in the limit of high asymmetry (α→∞). e maximal anticorrelationCov(G,P ) = −1/54β is realized at α? = 1+

√3 ≈ 2.73..., independent of β. e analytical curve is well reproduced

even for the moderate size n = 30 used in numerical simulations.

with mean and the covariance structure given by

E[Gn] = n(α+ 1)

(α+ 2), (7.20)

E[Pn] = n(α+ 1)2

(α+ 2)3, (7.21)

Var(Gn) =2

β

(α+ 1)2

(α+ 2)4, (7.22)

Var(Pn) =2

β

(α+ 1)2

(α+ 2)8

(α4 + 2α2 + 4α+ 2

), (7.23)

−Cov(Gn, Pn) =2

β

(α+ 1)2

(α+ 2)6α2 . (7.24)

e correlation coecient c(Gn, Pn) is

r(Gn, Pn) =Cov(Gn, Pn)√Var(Gn)Var(Pn)

= − α2

√α4 + 2α2 + 4α+ 2

. (7.25)

Proof. e joint asymptotically Gaussian behavior descends from the large deviation principle forthe empirical spectral measure µn of the β-ensemble with potential (7.19) and the contraction

105

principle jointly applied to the continuous maps

G[µn] :=

ˆdµ(x)x , P [µn] :=

ˆdµ(x)x(1− x) . (7.26)

Indeed, the joint cumulant κ`,m(Gn, Pn) of conductance and shot-noise is of orderO(n2−`−m)(see (4.16)) and then κ`,m(Gn, Pn) = o(1) for `+m > 2. is is sucient to establish the jointasymptotic normality of (Gn, Pn).

e average values E[Gn] and E[Pn] are given to leading order by

E[Gn] = n

ˆdλ %J (λ)λ , E[Pn] = n

ˆdλ %J (λ)λ(1− λ) . (7.27)

with %J (λ) in (7.16). e simplest way to evaluate the integralˆdλ %J (λ)a(λ) (7.28)

with a(λ) analytic is to introduce the complex valued function

f(z) =α+ 2

2πz

√z − λ−z − 1

, (7.29)

and consider the integralIε =

ˆΓε

dz f(z)a(z) , (7.30)

where the path of integration Γε (See Figure 7.4) in the complex plane is a clockwise contourwhich encloses the interval (the cut) [λ−, 1]. In the limit ε ↓ 0 we get

Re z

Im z

0 λ− 1

Γε

ε ε

Figure 7.4: Contour of integration in the complex plane for evaluating (7.30).

limε↓0

ˆγε

dz f(z)a(z) = 2i

ˆdλ %J (λ)a(λ) . (7.31)

On the other hand, if ε is suciently small, Iε = −2πi∑

zkRes (f(z)a(z); zk), where the sum of

the residues is over all the poles zk’s in the extended complex plane (in this case zk = 0 and thepoint at innity). Henceˆ

dλ %J (λ)a(λ) = −π [Res (f(z)a(z); 0) + Res (f(z)a(z);∞)] . (7.32)

106

Specializing the above formula to the functions a(x) = x and a(x) = x(1 − x) provides theaverage values in (7.20) and (7.21).

Concerning the covariance structure, in this case the conformal transformation is

x 7→ T (x) =λ−x+ 1

x+ 1, with λ− =

(α

α+ 2

)2

. (7.33)

Denoting a(x) = x and b(x) = x(1− x) we can compute for k > 0:

a(k) =

ˆR

dxλ−e

x + 1

ex + 1eikx =

iπ(λ− − 1)

sinhπk. (7.34)

b(k) =

ˆR

dxλ−e

x + 1

ex + 1

(1− λ−e

x + 1

ex + 1

)eikx =

iπ(λ− − 1)

sinhπk(λ−(k + i)− k) . (7.35)

A simple way to compute (7.34) and (7.35) is to consider

I(ω) =

ˆR

dxeiωx

ex + 1=

ˆR

dxei(ω+ i

2)x

2 cosh x2

, ω > 0 . (7.36)

Let consider the meromorphic function

f(z) =ei(ω+ i

2)z

2 cosh z2

(7.37)

in the complex plane. If ΓL is the rectangular contour in Fig. 7.5, by the residue theorem one has

Re z

Im z

0−L L

πi

2πiΓL

Figure 7.5: Contour of integration in the complex plane to evaluate (7.34).

ˆΓL

dz f(z) = 2πi Res (f(z);πi) . (7.38)

107

On the other hand, elementary calculus givesˆ

ΓL

dz f(z) =

ˆ L

−Ldz f(z) +

ˆ L+2πi

Ldz f(z) +

ˆ −L+2πi

L+2πidz f(z) +

ˆ −L−L+2πi

dz f(z)

=

ˆ L

−Ldx f(x) + i

ˆ 2πi

0dy f(iy) + e−2π(ω+ i

2)ˆ L

−Ldx f(x) + i

ˆ 0

2πidy f(iy) .

Hence, when L→ +∞ we get

limL→+∞

ˆΓL

dz f(z) = I(w)(

1 + e−2π(ω+ i2))

= 2πi Res (f(z);πi) . (7.39)

e residue can be evaluate by elementary methods

Res (f(z);πi) = limz→πi

ei(ω+ i2)z

sinh z2

= −e−πω (7.40)

Puing all together we nally get

I(ω) = − πi

sinhπω. (7.41)

Using this result it is immediate to get

a(k) =

ˆR

dxλ−e

x + 1

ex + 1eikx = λ−I(k + 1) + I(k) , (7.42)

that provides (7.34). A similar procedure provides (7.35). e covariance structure (7.22)-(7.23)-(7.24) readily follows from the covariance formula. For instance, the variance of the conductanceGn reads:

Var(Gn) =1

βπ2

ˆ +∞

0dk k tanh(πk)

π2(λ− − 1)2

sinh2(πk)

=(λ− − 1)2

β

ˆ +∞

0dk

2k

sinh(2πk). (7.43)

Using ˆ +∞

0dy

sin 2πay

sinh 2πy=

1

4tanh

πa

2, (7.44)

we nd by dierentiationˆ +∞

0dy

2y

sinh(2πy)=

d

da

ˆ +∞

0dy

sin 2πay

sinh 2πy

∣∣∣a=0

=1

8 cosh(0). (7.45)

en the result (7.22) is readily established using the relation (7.33) between λ− and α.

108

We have checked that this result is in agreement with the asymptotics of an exact nite-nexpression in [193] valid for all β (see also [159] for β = 1, 2, 4, [179] for β = 2 and α = 0, [130]for β = 1, 2 - and [171] for a dierent large-N method). e simple form (7.24) shows that forsuciently large n1,2 conductance and shot-noise are anticorrelated for any value of α ≥ 0. Fora symmetric (α = 0) cavity we recover the same covariance structure of Gn and Pn computedin Section 6.50. Given that their joint law is asymptotically Gaussian, they are also independentto leading order in n for α = 0. As shown in Fig. 7.3, at α? = (1 +

√3) (independent of β) the

anticorrelation between Gn and Pn is maximal and equal to Cov(Gn, Pn)∣∣α=α?

= −1/(54β).Since simultaneous measurement of conductance and shot noise are possible [210], a vericationof this “1 +

√3” eect might be within reach of current experimental capabilities.

As an additional bonus, from the joint limiting behavior of (Gn, Pn) we can deduce theconditional distribution of Gn given Pn = np and Pn given Gn = ng.

An interesting function of both Gn and Pn is the Fano factor Fn dened as

Fn =

∑i λi(1− λi)∑

i λi=PnGn

. (7.46)

We stress the fact that Fn is not a linear statistics of the transmission eigenvalues. e Fano factoris essentially the ratio of the actual shot-noise and the Poisson noise that would be measured ifthe system produced noise due to single independent electrons (for more physical details see thereview [27]). Since Gn and Pn are asymptotically jointly normal, we can use the Delta method(see for instance Section 5.5. of [227]) to conclude that Fn is asymptotically normal with

Fn ∼ AN (E[Fn],Var(Fn)) (7.47)

E[Fn] =E[Pn]

E[Gn]=

α+ 1

(α+ 2)2(7.48)

Var(Fn) =E[Pn]2

E[Gn]4Var(Gn)− 2

E[Pn]

E[Gn]3Cov(Gn, Pn) +

1

E[Gn]2Var(Pn)

=2

βn2

(α4 + 2α3 + 5α2 + 6α+ 3

)(2 + α)6

. (7.49)

In particular, for symmetric cavities (α = 0) we get

Fn ∼ AN(

1

4,

3

32βn2

). (7.50)

e average value E[Fn] of the Fano factor has been investigated in details in literature (see [192]).To the best of our knowledge the asymptotic distribution of the Fano factor was previouslyunknown and the computation of the variance (7.50) is new.

7.2.3 Joint distribution of traces of the β-ensembles

We address the following question: what is the behavior of Cov (TrHn,TrHm) as a function ofn and m for an invariant matrixH? Consider for simplicity ensembles whose %(λ) has supporton the interval σ = [0, λ+] (e.g. the Jacobi J or the Wishart-LaguerreW with c = 1).

109

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Β = 2

Λ+ = 4

0 5 10 15 2012

13

14

15

16

H2

ΠΒ

Var

HtrW

LL1

0 10 20 30 40 500.0

0.1

0.2

0.3

0.4

0.5

0.6

m

HΒ

ΠΛ

++m

LCov

HtrW

,tr

Wm

L

Figure 7.6: e covariance Cov(trW`, trWm) as a function ofm/n, whereW is a complex Wishart matrix (β = 2).For the simulation we sampled N = 104 Wigner-Laguerre matrices of size n = 800 and parameter c = 1. Main: Inthe simulation (circles) ` is xed ` = 50 and m varies from 1 to 50. e solid curve is (7.51). Inset: the numericalsimulations (squares) show the convergence of the rescaled variance to the limit value λ2

+ = 16.

Corollary 7.3. Consider a n-dimensional one-cut β-ensemble H specied by (7.1), whose densityof states has support on the interval σ = [0, λ+]. For large ` and m, both o(n)

Cov(

trH`, trHm)∼λ`+m+

βπ

√`m

`+m

[1 +O

(1

`m

)]. (7.51)

Seing ` = m, the following universal formula holds

lim`→∞

[2πβVar(trH`)

]1/`= λ2

+ . (7.52)

Proof. In this case the conformal map reads T (x) = λ+/(x+ 1). e deformed Fourier transform

110

of a(x) = x` is readily obtained aer a computation:

a(k) =

ˆR

dx eikx

(λ+

ex + 1

)`(7.53)

=

ˆ +∞

0dy yik−1

(λ+

y + 1

)`(7.54)(

with the position y + 1 =1

1− r

)(7.55)

= λ`+

ˆ 1

0dr rik−1 (1− r)`−1−ik (7.56)

= B (ik, `− ik) , (7.57)

where B(a, b) =´ 1

0 dt ta−1(1 − t)b−1 is Euler’s Beta function [10]. For large ` and m, we canuse the asymptotics B (ik, `− ik) = `−ik

(Γ(ik) +O(`−1)

). Using the covariance formula (7.3)

one computes to leading order in n:

Cov(

trH`, trHm)

=λ`+m+

βπ2

ˆ +∞

0dk k tanh (πk)Re [B (ik, `− ik) (B (ik,m− ik))∗]

∼λ`+m+

βπ2

ˆ +∞

0dk k tanh (πk) |Γ(ik)|2 cos

(k log

m

`

)=λ`+m+

βπ2I(

logm

`

)=λ`+m+

βπ

√`m

`+m. (7.58)

To derive the above result we have used the following representation of the Euler’s Gammafunction of an imaginary argument

Γ(ik) = (sin t− i cos t)

√π

k sinh(πk), (7.59)

witht = −γk +

∑j≥1

[k

j− arctan

k

j

](7.60)

where γ is the Euler’s constant. e above representation is quite useful to evaluate

I(θ) =

ˆ +∞

0dk k tanh (πk) |Γ(ik)|2 cos (k θ)

=

ˆ +∞

0dk

π

cosh (πk)cos (k θ)

= πsinh (θ/2)

sinh (θ), (7.61)

111

and get (7.58).

Formula (7.52) is in agreement with numerical simulations onW matrices (see Fig. 7.6). FortheW ensemble this limiting value is λ2

+ = 16 (see inset in Fig. 7.6), while in the J case we getλ+ = 1 recovering a result obtained in the context of the quantum transport problem (see Eq.(149) in [223] , and [172]).

e generalization to a generic connected and compact support is more intricate but can beworked out. We state here the exact result in `,m ≥ 1 (still, subleading terms in n are neglected).

Corollary 7.4. For a one-cut β-ensemble H whose density of states has support on the intervalσ = [λ−, λ+] the covariances of the spectral moments are given to leading term by

Cov(

trH`, trHm)

=λ`+m+

βπ2Γ(`)Γ(m)

∑j∈[`]

∑t∈[m]

(`

j

)(m

t

)(λ−λ+

)j+t׈ +∞

0dk k tanh(πk) Re [Γ(j + ik)Γ(`− j − ik)Γ(t− ik)Γ(m− t+ ik)]

=λ`+m+

βΓ(`)Γ(m)

∑j=0

m∑t=0

(`

j

)(m

t

)(λ−λ+

)j+tReA`mjt , (7.62)

where for `,m ≥ 1, 0 ≤ j ≤ ` and 0 ≤ k ≤ m we have dened

A`mjt =

ˆ +∞

0

dk

k sinh(2πk)

j−1∏i1=0

`−j−1∏i2=0

t−1∏i3=0

m−t−1∏i4=0

(i1 + ik)(i2 − ik)(i3 − ik)(i4 + ik) . (7.63)

Proof. A simple application of the covariance formula.

7.3 Derivation

We turn now to the derivation of the covariance formula (7.3).As we have already noted in Section 1.3.3, the average and the covariance ofAn =

∑ni=1 a(λi)

and Bn =∑n

i=1 b(λi) can be conveniently represented in terms of the average and covariance ofthe ESd. e mean value of An (or Bn) is

E[An] = n

ˆdλE [%n(λ)] a(λ) (7.64)

where E [%n] is the average spectral density. eir covariance is

Cov(An, Bn) = −n2

¨dλdλ′KN (λ, λ′)a(λ)b(λ′) , (7.65)

112

where the symmetric kernel

Kn(λ, λ′) = −Cov(%n(λ), %n(λ′)

)= −E

[%n(λ)%n(λ′)

]+ E [%n(λ)]E

[%N (λ′)

](7.66)

is the two-point (connected) correlation function.In order to describe the average and covariance in the large n limit, one should know both

the asymptotic behavior of E[%n] and Kn(λ, λ′). If we denote (provided they exist) by

%(λ) = limn→∞

E [%n(λ)] , (7.67)

K(λ, λ′) = limn→∞

n2Kn(λ, λ′) , (7.68)

the density of states and the two-point kernel respectively, assuming some regularity on a and b,we have for large n

E[An] = n

ˆdλ %(λ)a(λ) +O(1) , (7.69)

Cov(An, Bn) = −¨

dλdλ′K(λ, λ′)a(λ)b(λ′) +O(n−1) . (7.70)

e density of states %(λ) is the normalized minimizer of the energy functional

E [µ] = −ˆ

dµ(λ)dµ(λ′) log|λ− λ′|+ˆ

dµ(λ)V (λ) (7.71)

and, under the assumption of absolute continuity and single single support, it is the uniquesolution of the saddle-point equation

d

dλV (λ) =

%(λ′)

λ− λ′dλ′ , in λ : %(λ) > 0 . (7.72)

e two-point kernel K(λ, λ′) is provided by the following theorem.

eorem 7.5 (Functional derivative method [20]). For every n-dimensional β-ensemble

Kn(λ, λ′) =1

βn2

δE%n(λ)

δV (λ′)[1 + o(1)] . (7.73)

us, the two-point correlation kernel is

K(λ, λ′) = limn→∞

n2Kn(λ, λ′) =1

β

δ%(λ)

δV (λ′), (7.74)

the functional derivative of the density of states %(λ) with respect to the external potential V (λ′).

113

Proof. e average spectral density is nothing but the average of %n(λ) with respect to the jointlaw

Pβ(λ) =1

Zβe−βE(λ) . (7.75)

In the following computation of the variation of E%n(λ) with respect to the external potential wewill always neglect the subleading terms:

δE%n(λ)

δV (λ′)=

δ

δV (λ′)

ˆdλ %n(λ)Pβ(λ)

=δ

δV (λ′)

(1

Zβ

ˆdλ %n(λ)e−βn

2E[%n]

)=

(δ

δV (λ′)

1

Zβ

)ˆdλ %n(λ)e−βn

2E[%n] +1

Zβδ

δV (λ′)

ˆdλ %n(λ)e−βn

2E[%n]

= −βn2(E[%n(λ)%n(λ′)]−E[%n(λ)]E[%n(λ′)]

)(7.76)

where we have used

δE [%n]

δV (λ′)=

δ

δV (λ′)

[−1

2

¨%n(λ)%n(λ′′) log

∣∣λ− λ′′∣∣dλdλ′ +

ˆ%n(λ)V (λ)dλ

]= n2 %n(λ′) .

Remark 7.1. e limiting result (7.74) implies that, if limn→∞ V is independent on β, then thetwo-point correlation function depends on Dyson’s index as β−1.

Surprisingly enough, the two point correlation kernelK(λ, λ′) exhibits a stronger universalitythan %(λ) as shown by the following result. Again we denote σ = supp %.

eorem 7.6 (Beenakker-Brezin-Zee: bounded support). If σ = [λ−, λ+] is compact, the densityof states and the two-point kernel are

K(λ, λ′) =1

βπ2

1√(λ+ − λ)(λ− λ−)

d

dλ′

√(λ+ − λ′)(λ′ − λ−)

λ− λ′. (7.77)

Proof. is expression follows from the Tricomi formula (2.77). For a detailed proof see [36].

e above eorem shows explicitly that, while %(λ) depends on the details of the potentialthrough the Tricomi formula (2.77), the correlation kernel (7.77) depends on V only by the edgesof supp %. en, dierent Coulomb gases (matrix models), i.e. dierent potentials V , whoseequilibrium measure has the same support, have the same two-point correlation kernel. eexpression (7.77) also explains the covariance formula in real space (7.7) for σ bounded.

If the support of % is unbounded we have the following results.

114

eorem 7.7 (Dyson-Mehta: real line). If σ = R, the two-point kernel is

K(λ, λ′) = − 1

βπ2

d2

d(λ− λ′)2log|λ− λ′| . (7.78)

eorem 7.8 (Beenakker: positive half-line). If σ = [0,+∞), the two-point kernel is

K(λ, λ′) =1

βπ2

∂

∂λ

∂

∂λ′log

∣∣∣∣∣√λ−√λ′√

λ+√λ′

∣∣∣∣∣ . (7.79)

Proof. A derivation of both results can be found in [21]. For instance, using the functionalderivative method, establish (7.79) is equivalent to show that the right-hand side of (7.79) is thesolving kernel of the following singular integral equation

ˆ +∞

0dy %(y) log |x− y| = V (x) , x > 0 . (7.80)

We will derive this result in full details later.

Using conventional Fourier transform properties Dyson and Metha deduced from (7.78) theirvariance formula

Var(An) =1

βπ2

ˆ +∞

0dk k|a(k)|2 , (7.81)

with An =∑

i a(λi) and a(k) the Fourier transform of a(λ). is is the (DM) variance formula.irty years later, Beenakker provided a variance formula for linear statistics on the transmissioneigenvalues of a chaotic cavity. e main obstruction he succeeded to overcome is that the kernel(7.79) is not translationally invariant. Ingeniously, he used the change of variables λ = ex andhe could derive a formula in the context of quantum transport (what we called the (B) varianceformula):

Var(An) =1

βπ2

ˆ +∞

0dk k tanh(πk)|a(k)|2 , (7.82)

where now a(k) is the Mellin transform of a(ex) (see more below).ite surprisingly, for many years these have been the only variance formulas for β-ensembles.

7.3.1 Some lemmas on conformal maps and Coulomb gas systems

e next few lemmas reveal that conformal maps interact well with the logarithmic interaction.Essentially we will see that a conformal change of coordinates, transforms a 2D Coulomb gassystem into another 2D Coulomb gas, i.e. the interaction is preserved. More surprisingly is thenontrivial fact that the external temperature β−1 does not change by applying this family oftransformations. e consequences of this nice fact is what we call “conformal map method”: bya suitable change of variables, a generic Coulomb gas system can be mapped to a reference onewhich is solvable. Pulling back this transformation we eortless extract the essential properties

115

of the original Coulomb gas system. We will say that λ = (λ1, . . . , λn) ∈ Rn is a Coulomb gassystem at inverse temperature β (in the external potential V : R→ (−∞,+∞]), if λ is a randomvector distributed according to

law(λ) =1

Ze−βE(λ) , (7.83)

E(λ) = −∑i<j

log |λi − λj |+ n∑i

V (λi) +O(n) . (7.84)

e measure (7.83) is the Gibbs-Boltzmann measure associated to a 2D Coulomb gas connedon a line with a conning potential V (λ) at temperature β−1. We will always assume V suchthat in the thermodynamic limit (n→∞) the average spectral density E%n(λ) converges to a(nonrandom) density of states %(λ) on the line supported on a single interval.

For a, b, c, d ∈ R with ad− bc 6= 0, let us consider the conformal map

λ = T (x) : λi =axi + b

cxi + d∀i ∈ [n]

(x = T−1(λ) : xi =

b− dλicλi − a

)(7.85)

e following lemmas will clarify why conformal maps interact nicely with Coulomb gas systems.

Lemma 7.9. If λ is a Coulomb gas system at temperature β−1, then x = T−1(λ) is a Coulombgas system at the same temperature.

Proof. e law of x is the pushforward of law(λ) by the conformal map T :

law(x) =1

Ze−βE(Tx)

∣∣∣∣det

(∂λi∂xj

)∣∣∣∣ (7.86)

=1

Ze−βE(Tx)+

∑i,j log

[ad−bc

(cxi+d)2

]δij

(7.87)

=1

Ze−βE(Tx)+O(n) . (7.88)

Let us evaluate the energy function

E(Tx) = −∑i<j

log

∣∣∣∣axi + b

cxi + d− axj + b

cxj + d

∣∣∣∣+ n∑i

V

(axi + b

cxi + d

)(7.89)

= −∑i<j

log

∣∣∣∣(ad− bc)(xi − xj)(cxi + d)(cxj + d)

∣∣∣∣+ n∑i

V

(axi + b

cxi + d

)(7.90)

= −∑i<j

log |xi − xj |+ n∑i

V

(axi + b

cxi + d

)(7.91)

+∑i<j

log |(cxi + d)(cxj + d)| − n(n− 1)

2log |ad− bc| . (7.92)

116

en

law(x) =1

Ze−βE

T (x) , (7.93)

ET (x) = −∑i<j

log |xi − xj |+ n∑i

V T (xi) +O(n) . (7.94)

ZT = Ze−βn(n−1)

2log |ad−bc| (7.95)

V T (x) = V

(ax+ b

cx+ d

)+ log |cx+ d| (7.96)

Remark 7.2. is Lemma is the main ingredient we are going to use. e statement of the Lemmais two-fold: i.) the logarithmic interaction is conformal invariant and ii.) temperature does notchange under this family of transformations! is second point is crucial and is absolutely notobvious a priori.

A conformal change of variables modies the density of states of the systems in the followingway.

Lemma 7.10. Let %(λ) be the density of states of λ. en the density of states %T (x) of x is

%T (x) = %

(ax+ b

cx+ d

)ad− bc

(cx+ d)2. (7.97)

Proof. e change of variables λ = ax+bcx+d gives

%T (x) = %(λ(x))dλ

dx. (7.98)

is Lemma 7.10 shows how a particular a change of coordinated T modify the support of %.Here we recall that an interval (a connected subset of R) is mapped into another interval by aconformal map.

Lemma 7.11. Let %(λ) be the density of states of λ and suppose that its support σ = [λ−, λ+] isconnected, with λ− < λ+ (nite or not). Let us exclude the case σ = R. en, if λ = Tx witha/c = λ− and b/d = λ+, the support σT of %T is the positive half-line σT = [0,+∞).

Proof. A trivial application of Lemma 7.10.

Remark 7.3. Here is a good time to x ideas. If λ is a Coulomb system at temperature β−1

with density of states supported on [λ−, λ+], then x dened by λi = axi+bcxi+d

(ad− bc 6= 0) witha/c = λ− and b/d = λ+ is a Coulomb gas at the same temperature whose density of states %Thas support σT = [0,+∞). Explicitly we can choose the following transformations:

117

• If supp% = [λ−, λ+], then λi = λ−xi+λ+xi+1 ;

• If supp% = [λ−,+∞), then λi = λ− + 1xi

;

• If supp% = (−∞, λ+], then λi = −xi + λ+.

No conformal transformation can map R onto the positive half-line. en the case supp % = Rcannot be handled with this technique.

7.3.2 Solving kernel and universal covariance formula

As promised we derive the two-point kernel of eorem 7.8 . is is equivalent to prove thefollowing result.

Lemma 7.12. e solving kernel ofˆ +∞

0dy %(y) log |x− y| = V (x) , x > 0 (7.99)

is given by

Φ(x, x′) =1

π2

d

dx

d

dx′log

∣∣∣∣∣√x−√x′

√x+√x′

∣∣∣∣∣ . (7.100)

In the proof of this lemma we will use some properties of the Mellin transform. We recall herethe denition and some useful properties [59, 134]. e Mellin transform of an integrable realfunction f , when it makes sense, is dened as

Mf(x); s =

ˆ +∞

0dxxs−1 f(x) s ∈ C . (7.101)

Once f is integrable, the existence of (7.101) only depends on f decreasing fast enough at 0and +∞. In particular, if f(x) = o(xα) for x → 0 and f(x) = o(xβ) for x → ∞, thenMf(x); s exists for any s ∈ C such that Re s ∈ (−α,−β). We list here the properties of theMellin transform that we are going to use. eir proof follows from the properties of the Fouriertransform since the Mellin transform of f(x) is nothing but the Fourier transform of f(ex).

(i) (Translation)Mxpf(x); s =Mf(x); s+ p , p > 0 ; (7.102)

(ii) (Multiplicative convolution)

Mf(x); sMg(x); s =M(f ? g)(x); s

=Mˆ +∞

0

dy

yf

(x

y

)g(y); s

; (7.103)

118

(iii) (Dilation operator)

M(x∂x)n f(x); s = (−s)nMf(x); s ; (7.104)

Proof of Lemma 7.12 . e singular integral equation with the logarithmic kernelˆ +∞

0dy %(y) log |x− y| = V (x) , x > 0 (7.105)

can be conveniently wrien asˆ +∞

0dy %(y) log

∣∣∣∣x− yy∣∣∣∣ = f(x) , x > 0 (7.106)

where f(x) = V (x)− V (0). Applying the Mellin transform on both sides we get

Mˆ +∞

0dy %(y) log

∣∣∣∣x− yy∣∣∣∣; s =Mf(x); s . (7.107)

We evaluate now the le hand side of the above equation:

Mˆ +∞

0dy %(y) log

∣∣∣∣x− yy∣∣∣∣; s =

¨ +∞

0dy dx %(y)xs−1 log

∣∣∣∣xy − 1

∣∣∣∣ (7.108)

=

ˆ +∞

0dy %(y)

π

s

ys

tan (πs)(7.109)

=π

s

1

tan (πs)

ˆ +∞

0dy ys%(y) (7.110)

=π

s

1

tan (πs)M%(y); s+ 1 . (7.111)

Combining (7.107) and (7.111) we get

Mx%(x); s =s

πtan (πs)Mf(x); s (7.112)

=M(ˆ +∞

0

dy

yM−1

s

πtan (πs);

x

y

f(y)

); s

(7.113)

where the rst equality follows trivially from (7.111) by using the translation property (7.102) ofthe Mellin transform, and in (7.113) we have used the multiplicative convolution theorem (7.103).Now, by inverting both sides of the above equality and using the dilation property (7.104)

M−1 s tan (πs);x =M−1

s2 tan (πs)

s;x

= (x∂x)2M−1

tan (πs)

s;x

=

1

π(x∂x)2 log

∣∣∣∣1 +√x

1−√x

∣∣∣∣ (7.114)

119

we get

x%(x) =

ˆ +∞

0

dy

yM−1

s

πtan (πs);

x

y

f(y) (7.115)

=1

π2

ˆ +∞

0

dy

y

[x

y

∂

∂ (x/y)

x

y

∂

∂ (x/y)log

∣∣∣∣√y +√x

√y −√x

∣∣∣∣] f(y) , (7.116)

and nally, using the chain rule, we recover the solving kernel (7.100):

%(x) =1

π2

ˆ +∞

0∂x∂x′ log

∣∣∣∣∣√x+√x′

√x−√x′

∣∣∣∣∣ f(x′)dx′ . (7.117)

Lemma 7.13. Let x be a Coulomb gas system whose density of states is supported on the positivehalf-line. en, for any dierentiable linear statistics An =

∑i a(xi) and Bn =

∑i b(xi):

Cov(A,B) =1

βπ2

ˆ +∞

0dk k tanh (πk)Re [Ma(x); (k)Mb(x); (k)∗] (7.118)

where we have introduced the Mellin transform

Mf(x); (k) =

ˆ +∞

−∞dx eikxf(ex) (7.119)

and we have neglected subleading terms.

Proof. From eorem 7.5 the two-point correlation function is given by

K(λ, λ′) =1

β

δ%(λ)

δV (λ′), (7.120)

while the equilibrium condition of the density of states readsˆR

dλ′ %(λ′) log|λ− λ′| = V (λ) , for λ > 0 . (7.121)

From Lemma 7.12 the solving kernel of (7.121) (recall that the support of % is the positive half-line)is given by (7.100). en, under the smoothness condition of a and b:

Cov(A,B) =1

βπ2

¨ +∞

0dλdλ′ log

∣∣∣∣∣√λ−√λ′√

λ+√λ′

∣∣∣∣∣da(λ)

dλ

db(λ)

dλ′. (7.122)

Here it is clear the reason of the smoothness assumptions on the linear statistics. As argued in [20],the change of variable x = log y makes the kernel translation invariant and (7.123) becomes

Cov(A,B) = − 1

βπ2

ˆ +∞

0dx dx′ a(ex)b(ex

′)

d2

d (x− x′)2 log

∣∣∣∣tanhx− x′

4

∣∣∣∣ . (7.123)

120

In this form, we can change in Fourier coordinates and using the (Fourier) convolution theorem.e covariance formula (7.3) follows using the Parseval identity, the standard properties of theFourier transform and the identity

ˆ +∞

−∞dx eikx log

∣∣∣tanhx

4

∣∣∣ = −√

2π3 tanh (kπ)

k. (7.124)

Now we can state our main result.

eorem 7.14. Let λ be a Coulomb gas system whose density of states has connected support σand let An =

∑i a(xi) and Bn =

∑i b(xi) two linear statistics on λ. en, with the notation of

(7.5)-(7.6), if (7.3) converges then

Cov(An, Bn)→ 1

βπ2

ˆ ∞0

dk ϕ(k) Re[a(k)b∗(k)] , as n→∞ . (7.125)

Proof. In the case σ = R the covariance formula (7.125) is an immediate extension of the (DM)variance formula for two linear statistics. Let us suppose σ 6= R. In this case, by Lemmas 7.9,7.10 and 7.10 the conformal map T dened in (7.6) transforms the Coulomb gas λ at inversetemperature β in the Coulomb gas x at the same temperature, whose density of states is supportedon the whole positive half-line. For this new Coulomb gas, since %T is supported on the positivehalf-line, the two-point correlation kernel KT (x, x′) is given by (7.100). Formula (7.125) is thenan immediate application of the previous Lemma 7.13.

We conclude with few remarks. e point we want to point the aention here is that ourderivation is unied and perhaps claries some confusing statements in the earlier literature. Forexample, a persisting ambiguity has gone unnoticed between the range Λ of the eigenvalues andthe support σ of the limit spectral density. We resolve this ambiguity, introducing the conformalmap method that encloses alle cases in a unied framework. e main usefulness of the conformalmap method (for σ 6= R) is evident: the asymptotic kernel KT (x, x′) of the transformed gasbecomes universal (independent of details of the potential V (x) and even of the edge points λ±of the original density %(λ)), yielding the xed kernel ϕ(k) in (7.3). Every surviving trace of theoriginal ensemble is encoded in λ±, which have now been moved inside the argument of thelinear statistics. ere is however an important exceptional case: Coulomb gases whose densityof states invades the whole real line. is class is separated from the log-gases with conneddensity of states, and this explains the asymmetry of the covariance formula. We believe thatthis polishing operation should be per se valuable. From a pragmatic point of view, we shouldmention that both the variance formulas (DM) and (B) have been almost ignored for applications.We are aware only of the use of (B) to derive uctuation formulas for symmetric cavities. For thisreason we preferred to provide an exhaustive list of applications of our covariance formula.

e range of validity of the covariance formula is still not completely clear. We stated that theconvergence of (7.3) is the optimal condition but it is not simple to predict when this is the case.

121

We can however combine our knowledge on the Coulomb gases to state a necessary condition.Let be An = nA[%n] be a linear statistics. As seen in Chapter 4, the uctuations around the meanare detected by the behavior of the saddle-point density of the variational problem with energyfunctional

Es[µ] = −ˆ

dµ(λ)dµ(λ′) log|λ− λ′|+ V [µ] + sA[µ] . (7.126)

Here s is the Laplace variable associated to the random variable An. en, a necessary conditionto apply the variance (or covariance) formula is that there exists a neighbourhood around s = 0such that the limiting spectral density is still supported on a single interval, i.e. there shouldexists ε > 0 such that for all |s| < ε, supp %?s is an interval (the one-cut Assumption 7.1 should berobust for suciently small perturbations).

Part III

BEYOND THE CLASSICALENSEMBLES

122

Chapter 8

e Unbiased Ensemble

is Part is devoted to the application of Random Matrix eory in antum Information. Wewill introduce the unbiased ensemble and in the following chapters we will move beyond theclassical ensembles.

e rst part of this Chapter will serve to set the notation and highlight the main conceptsthat we will need. In particular we will dene and briey discuss entanglement and how toquantify it. Hereaer the underlying eld is C.

C∗-algebras are the mathematical models for quantum systems, and subsystem are simply∗-subalgebras with unit. States (normalized, positive linear functional on these algebras) describethe conguration of the system. Clearly, the set of states of a system is convex. e extremalpoints of this convex set are called pure states. In constrast, all others are named mixed. Anynormal state ω can be uniquely represented as ω(A) = tr (ωA) for some positive element ω ∈ Awith trace 1 (these are called density operators).

By the celebrated Gelfand-Naimark-Segal (GNS) construction, every C∗-algebra is isomorphicto a ∗-subalgebra of some B(H ), the algebra of bounded linear operators on the Hilbert spaceH . In the following we will always deal with nite-dimensional algebras. is will alleviate usfrom various technical issues. In this case one can consider H ' Cd for some integer d ≥ 2. Letus give some basic denitions in the nite-dimensional seing.

8.1 A short quantum-mechanical vocabulary

In this section we will present the minimal vocabulary of quantum mechanics and quantuminformation. We refer to the literature instead of giving full proofs. Besides the classical treatiseswe recommend the recent books of Bengtsson and Zyczkowski [24] and Holevo [111].Denition 8.1 (antum states). A quantum state (or density operator) is a positive (Hermitian)matrix ω with trω = 1. A states ω is pure if ω2 = ω, all others are named mixed. e set ofdensity operators acting on H is denoted as D(H ).

Pure states are rank-1 projection operators. In the Dirac notation a pure state is denotedω = |ψ〉 〈ψ| for some normalized |ψ〉 (note that the embedding of the unit sphere onto the pure

123

124

state space |ψ〉 ∈ S(H ) 7→ |ψ〉 〈ψ| ∈ D(H ) is U(1)-gauge invariant). Mixed states are nontrivial convex combinations of pure states. Among all mixed states we distinguish the maximallymixed state ω = 1/ dim H . Tensor product Hilbert spaces H = HA ⊗HB describe bipartitequantum systems. Given a state ω ∈ D(HA ⊗HB), the states ωA = trB ω and ωB = trA ωare called reduced density operators on HA and HB , respectively. e maps trA and trB arepartial traces. Hereaer we will always denote dA = dim HA and dB = dim HB . Most of thedenitions and results we quote can be found in any standard material of quantum information,for instance see [228].

Denition 8.2 (Entanglement). A state ω is separable with respect to the bipartition (HA,HB)if for some integer k ≥ 1

ω =∑i∈[k]

piω(i)A ⊗ ω

(i)B (8.1)

where the pk’s are probability weights (pi ≥ 0 and∑

i∈[k] pi = 1) and the ω(i)A ’s (resp. ω(i)

B ) arestates of HA (resp. HB).

For pure states the denition of separability simplies. Indeed a pure state |ψ〉 〈ψ| is separableif |ψ〉 is a decomposable element of HA ⊗HB , i.e. |ψ〉 = |u〉 ⊗ |v〉 for some |u〉 ∈ HA and|v〉 ∈HB .

Lemma 8.1 (Schmidt decomposition). Every pure state |ψ〉 ∈HA ⊗HB (suppose dA ≤ dB) canbe wrien as

|ψ〉 =∑i∈[dA]

√λi |ui〉 ⊗ |vi〉 , (8.2)

with Schmidt coecients λi ≥ 0,∑

i∈[dA] λi = 1 and |ui〉, |vi〉 orthonormal sets of HA andHB , respectively. e number r ∈ [min(dA, dB)] of nonzero Schmidt coecients is called Schmidtnumber of |ψ〉. Clearly, a pure state is separable i r = 1. e Schmidt coecients λi’s of |ψ〉 arethe eigenvalues of ωA (and ωB).

A maximally entangled state is a state |ψ〉 with r = min(dA, dB) and Schmidt coecients allequal to 1/r.

ere is a whole zoo of entanglement measures. We refer the reader to [48, 112]. Here we willconsider entanglement measures for pure states (of bipartite systems), a restriction that notablysimplies the task of entanglement quantication.

Denition 8.3 (Renyi’s entropies). For all q ≥ 0 the q-Renyi’s entropy of a state ω is dened as

Sq(ω) =1

1− qlog trωq . (8.3)

e limit of the family Sq(ω) for q → 1 exists and is called von Neumann entropy:

SvN(ω) = limq→1

Sq = −tr ω logω . (8.4)

125

e other limits q →∞ and q ↓ 0 corresponds respectively to the so-called Chebychev entropy

S∞(ω) = limq→∞

Sq(ω) = − log ‖ω‖ , (8.5)

and the rank entropyS0(ω) = lim

q→0Sq(ω) = log rankω . (8.6)

e linearized entropy Σq , for q ≥ 0 are simply dened by the equation

Sq(ω) =1

1− qlog Σq(ω) (8.7)

e Renyi’s entropies are normalized in a uniform way, in the sense that for all q ≥ 0:0 ≤ Sq ≤ log d. e lower bound is aained on pure states Sq(|ψ〉 〈ψ|) = 0; the upper boundis saturated by the mixed state Sq(1/d) = log d. e linearized Renyi’s entropies are dened asΣq(ω) = trωq . For q = 2 the name purity is quite common in the quantum information literature.

Lemma 8.2. A pure state |ψ〉 ∈HA⊗HB is separable i for some q ≥ 1 (and then for all q ≥ 1)Sq(trB |ψ〉 〈ψ|) = 0. A pure state |ψ〉 ∈ HA ⊗HB is maximally entangled i for some q ≥ 1(and then for all q ≥ 1) Sq(trB |ψ〉 〈ψ|) is maximum.

For later convenience we recall here the foliation of the pure states sphere of a bipartite systemin disjoint orbits of local unitaries [200].

Denition 8.4 (Local unitaries). Let H = HA⊗HB be a bipartite Hilbert space. e set of uni-tary operators UA ⊗ UB : UA ∈ U(HA) andUB ∈ U(HB) is a group under the compositionof operators. is group is isomorphic to U(HA)× U(HB), and will be denoted in this way. Wecall U(HA)× U(HB) the group of local unitary transformations. e orbit of U(HA)× U(HB)through a pure state |ψ〉 ∈HA ⊗HB is called the local orbit through |ψ〉.

Observe that U(HA)× U(HB) does not act transitively on the sphere of pure states. Indeed,quantum correlations between two systems cannot be modied by unitary transformationsperformed locally on each subsystem. Given a pure state |ψ〉 of a bipartite quantum systemHA ⊗HB , any state belonging to the same local orbit of U(HA)× U(HB) through |ψ〉 has thesame set of Schmidt coecients of |ψ〉. en, the local unitary transformations induce a nicefoliation of the space of pure states: a foliation in manifolds of equal entanglement. e dimensionof the orbit of the local unitaries UA⊗UB through |ψ〉 depends on the degeneracy of the Schmidtcoecients of |ψ〉.

For simplicity we will assume in the following dA ≤ dB . Let us consider the surjective maps (whose existence is ensured by the Schmidt decomposition) that maps a pure state into theordered dA-ple of its Schmidt coecients

s : S(HA ⊗HB) 3 |ψ〉 7→ (λ1, . . . , λdA) ∈WdA , (8.8)

126

where the n-dimensional Weyl chamber Wn (for n ≥ 1) is dened as

Wn =

λ ∈ Rn : 0 ≤ λ1 ≤ λ2 ≤ · · · ≤ λn ,∑i∈[n]

λi = 1

. (8.9)

e coordinates of s(|ψ〉) = (λ1, . . . , λdA) in the Weyl chamber are the Schmidt coecients of|ψ〉 in nondecreasing order. By xing the degeneracy of the Schmidt coecients we x a face inthe Weyl chamber. In order to describe degeneracy let us write the Schmidt eigenvalues as

s(|ψ〉) =(0, . . . , 0, ν1, . . . , ν1, ν2, . . . , ν2, . . . , νk, . . . , νk

), (8.10)

where 0 < νi < νj for i 6= j. Each νj occurs mj times, and m0 is the number of zeros of s(|ψ〉)(the number of vanishing Schmidt coecients). We will call the set of integers

m(|ψ〉) = (m0,m1, . . . ,mk) (8.11)

the degeneracy of the Schmidt coecients of |ψ〉. It is clear that there is a one-to-one corre-spondence between “faces” of the Weyl chamber and partitionsm = (m0,m1, . . . ,mk) ∈ πdAof the integer dA, i.e. m0 +

∑i∈[k]mi = dA. If we denote by ‖m‖0 the number of nonzero

components of m = m(|ψ〉), the number of distinct nonzero Schmidt eigenvalues of |ψ〉 isk = ‖m‖0 −m0. Let us denote by Wm the portion of the chamber with degeneracy labelledby m. Clearly WdA =

⋃m∈πdA

Wm. e preimage by s of a point x ∈ Wm is the local orbitU(HA) × U(HB) |ψ〉 ≡ O|ψ〉 (= s−1(x)) of some pure state |ψ〉. e dimension of this localorbit depends only on m(|ψ〉). If |ψ〉 is separable or maximally entangled we will denote thelocal orbits byOSep andOME, respectively. In both cases the number of nonzero distinct Schmidteigenvalues if k = 1.

A complete analysis of local orbits has been performed by Sinolecka, Zyczkowski and Kus[200] in the simplied case dA = dB .eorem 8.3 (Sinolecka, Zyczkowski and Kus [200]). Let |ψ〉 ∈ S(HA ⊗HB) with dA = dB .Let m(|ψ〉) = (m0,m1, . . . ,mk) ∈ πdA the degeneracy of s(|ψ〉). en the local orbit O|ψ〉 hasdimension

dimO|ψ〉 = dim s−1(x) = 2d2A − 2m2

0 −∑i∈[k]

m2i − 1 . (8.12)

In particular we have

(Generic states) If k = dA, i.e. m(|ψ〉) = (0, 1, . . . , 1), then

dimO|ψ〉 = 2d2A − dA − 1 ; (8.13)

(Separable states) If O|ψ〉 = OSep, i.e. m(|ψ〉) = (dA − 1, 1), then

dimOSep = 4(dA − 1) ; (8.14)

(Maximally entangled states) If O|ψ〉 = OME, i.e. m(|ψ〉) = (0, dA), then

dimOME = d2A − 1 . (8.15)

127

8.2 Typicality for the unbiased ensemble

In the last years many researchers have been investigating the typical properties of random purestates, i.e. unit vectors drawn at random from the Hilbert space associated to a quantum system.is subject has aracted the aention in several directions, and some important results havebeen achieved mostly dealing with the characterization of entanglement [239, 196, 240, 109, 81,102, 103, 82, 146, 44, 149, 163, 132, 53].

e standard ensemble which has been intensively investigated is that of random pure stateswith probability law induced by the Haar measure on the unitary group. e sphere of pure statesS(H ) is an orbit of the unitary group U(H ). en, the normalized Haar measure on the groupinduces a (unique) unitarily invariant probability measure on the sphere of pure states.

Denition 8.5 (Unbiased ensemble). By a pure state |ψ〉 ∈H drawn from the unbiased ensemblewe mean a random unit vector distributed according to the unitarily invariant measure on S(H ).We will use the notation |ψ〉 ∼ Unif(H ).

is ensemble, being the maximally symmetric one, implements in a natural way the caseof minimal knowledge of a quantum state [106]. It is structureless, in the sense that the inducedmeasure only depends on the dimension of the total Hilbert space and it is not sensitive to anytensor product structure [239, 1].

For these reasons, the unitarily invariant ensemble is known as the unbiased ensemble [53].e typicality for this ensemble has been deeply investigated by exploiting the large technologyprovided by Random Matrix eory. Here we recollect some basic facts about the entanglementproperties of the unbiased ensemble. From now on we will be considering a random pure state|ψ〉 ∼ Unif(HA ⊗HB) with dA ≤ dB . Its reduce state on HA is denoted by ωA = trB |ψ〉 〈ψ|.In this thesis we will mostly use the local purity (see (8.7))

πAB(|ψ〉) = Σ2(ωA) ∈[

1

dA, 1

](8.16)

as a measure of separability of pure states. e purity is maximum πAB = 1 on the set of separablestates and aains its minimal value πAB = 1/dA on maximally entangled states. e rst resultsabout the entanglement distribution in the unbiased ensemble have been computation of averagevalues:

Lemma 8.4. Let |ψ〉 ∼ Unif(HA ⊗HB). en

E[πAB(|ψ〉)] =dA + dBdAdB + 1

, (8.17)

and

E[SvN(ωA)] =

dAdB∑k=dB+1

1

k− dA − 1

2dB. (8.18)

128

e average value of the local purity of a random pure state (8.17) was computed usinga diagrammatic method by Lubkin [147]. e average local von Neumann entropy (8.18) wasconjectured by Page in his pioneering work [180] and proved in [97]. See also [198]. eexpectation values (8.17)-(8.18) are also the typical values of the local purity and the von Neumannentropy thanks to the concentration of measure phenomenon.

Lemma 8.5 (Levy’s lemma (adaptation from [110])). Let f : S(H )→ R be a Lipschitz functionon the unit sphere of H (dim H = d) with Lipschitz constant η > 0. en, if |ψ〉 ∼ Unif(H ):

Pr |f(|ψ〉)−E [f(|ψ〉)]| > ε ≤ 2 exp

(− 2dε2

9π3η2

). (8.19)

e Levy’s lemma is a concentration result for random vectors uniformly distributed onhigh-dimensional spheres. It is a well-known fact that in large dimensions the uniform measureon the sphere concentrates about any equator. is means that any polar cap properly containedin a hemisphere has relative volume exponentially small in the dimension of the sphere. Asimple manipulation shows that the same result holds for slowly varying functions (e.g. Lipschitzfunction). Any slowly varying random variable induced from the uniform measure on the sphere inconcentrated about its mean value, in the sense that deviations from the average are exponentiallysuppressed as the dimension of the sphere increases. en, given a Lipschitz function f(|ψ〉), therecipy to apply Levy’s lemma in order to show the typicality of E[f ] is the following

(i) Compute (or estimate) E[f ];

(ii) Compute (or estimate from above) the Lipschitz constant η of f ;

(iii) Plug in the large deviation inequality (8.19).

Using this recipe, one can easily show the concentration of measure for the local purity and vonNeumann entropy. Indeed, the local purity has Lipschitz constant bounded by η ≤ 4, while forthe von Neumann entropy η ≤

√8 log dA, for dA ≥ 3 (see [110] for details). en, by virtue of

Levy’s Lemma 8.5:

Pr |πAB(|ψ〉)−E [πAB]| > ε ≤ 2 exp(−C1dAdBε

2)

(8.20)

Pr |SvN(ωA)−E [SvN]| > ε ≤ 2 exp

(−C2dAdBε

2

(log dA)2

), (8.21)

where the absolute constants can be choosen to be C1 = (72π3)−1 and C2 = (36π3)−1.In fact, Lemma 8.5 is the counterpart of a more classical result for independent Gaussian

variables (see [211] for a proof):

Lemma 8.6. LetX = (X1, X2, . . . , Xk) be a vector with independent identically distributed Gaus-sian components, with distribution Xi ∼ N (0, σ2). en, for any f : Rk → R, with Lipschitzconstant η > 0, the following concentration inequality holds

Pr |f(X)−E[f(X)]| > ε ≤ 2 exp

(− ε2

4η2σ2

). (8.22)

129

is lemma can be converted for our purposes as

Lemma 8.7. Let φ = (φ1, φ2, . . . , φd) be a random vector in H with independent identicallydistributed Gaussian components, with distribution φi ∼ NC(0, 1/d). en, for any f : H → R,with Lipschitz constant η > 0, the following concentration inequality holds

Pr |f(φ)−E[f(φ)]| > ε ≤ 2 exp

(− dε

2

2η2

). (8.23)

A random vector φ = (φ1, φ2, . . . , φd) in H with independent identically distributed compo-nents φi ∼ NC(0, 1/d) is exponentially close to the unit sphere. is can be easily seen by virtueof Levy’s lemma. Another way to see it, is by realizing that, due to the rotational invariance of theGaussian law, φ =

√r |ψ〉 where |ψ〉 ∼ Unif(H ) is uniformly distributed on the sphere S(H )

while r ∼ χ22d/2d where χ2

2d is a chi-square random variable with 2d degrees of freedom (i.e. thesum of squares of 2d independent standard Gaussian variables). A standard computation (see forinstance Section 7.5 of [231]) gives for t ≤ d:

E[etr] =1

(1 + t/d)d. (8.24)

en by Markov inequality we get

Pr (r ≥ 1 + ε) ≤ e−t(1+ε)

(1 + t/d)d. (8.25)

Optimize this bound, we set t = dε/(1 + ε) and get for all 0 < ε < 1:

Pr (r ≥ 1 + ε) ≤ e−d[ε−log(1+ε)] ≤ e−dε2

4 . (8.26)

Similarly we can establish the bound

Pr (r ≥ 1− ε) ≤ ed[ε+log(1−ε)] ≤ e−dε2

4 . (8.27)

Finally we get that the probability that r deviates from 1 is exponentially suppressed as

Pr (|r − 1| ≥ ε) ≤ 2e−dε2

4 . (8.28)

is shows that the Gaussian ensemble φ is a normalized vector with overwhelming probabilityand justify the Gaussian approximations that we are going to use several times. Moreover thepurity of the Gaussian state φ is given by

πAB(φ) = r2πAB(|ψ〉) . (8.29)

en

Pr (|πAB(φ)− πAB(|ψ〉)| ≥ ε) = Pr(πAB(|ψ〉)|r2 − 1| ≥ ε

)≤ Pr

(|r2 − 1| ≥ ε

), (8.30)

130

and a similar argument as before can be used to bound these deviations.Another useful trick borrowed from high-dimensional geometry is to combine a large deviation

bound due to a concentration of measure phenomenon with an union bound over an ε-net. Werecall here a standard denition [128].

Denition 8.6 (ε-net). e set Aε ⊂ Rn is said to be an ε-net for A ⊂ Rn if, for every x ∈ A,there exist at least one point y ∈ Aε such that ‖x− y‖ ≤ ε.

e above denition can be extended to any metric space. It is easy to show that the nite-dimensional sphere S(H ) is totally bounded, i.e. for every ε > 0 there exists a nite ε-net Sεfor S(H ). In particular, for 0 < ε < 1 it is easy to bound the cardinality of these ε-nets as|Sε| ≤ (5/ε)2d (see Lemma III.6 of [110]). Suppose that we have established a large deviationbound for a Lipschitz function f on the sphere S(H ). is amounts to say that for every|ψ〉 ∈ S(H ) the probability that f(|ψ〉) deviates from E[f ] more than δ is extremely small, say

Pr (|f(|ψ〉)−E[f(|ψ〉)]| > δ) ≤ γ . (8.31)

If we consider an ε-net Sε, the probability that f(|ϕ〉) deviates by an amount δ from its averageon every point |ϕ〉 ∈ Sε is controlled as

Pr(for all |ϕ〉 ∈ Sε : |f(|ϕ〉)−E[f ]| > δ) ≤ |Sε|η . (8.32)

en, by balancing the cardinality of the ε-net and the large deviation bound, we can convertthese probabilistic statements into a deterministic one saying that |f(|ψ〉) − E[f ]| ≤ δ + ηεeverywhere.

8.3 Random matrix theory of the unbiased ensemble

e information about entanglement of a pure state |ψ〉 ∈HA⊗HB is encoded in the spectrum ofthe reduced density operator ωA ∈ D(HA). en, when dealing with random pure states, as longas we are concerned with entanglement properties, we would like to trace out all the randomnessbut the statistical properties of the Schmidt eigenvalues λii∈[dA] of ωA. Remarkably, this ispossible. Indeed the pushforward of the unbiased ensemble of pure states |ψ〉 ∼ Unif(HA⊗HB)by the partial trace trB (·) is the ensemble of random density matrices ωA = trB (|ψ〉 〈ψ|) ∈D(HA). A standard matrix theory argument (see [24, 239, 240]) predicts that ωA is a unitarilyinvariant matrix model and the law of its eigenvalues λ = (λ1, . . . , λdA) is explicitly known [145]

PdA,dB (λ) =1

ZdA,dB

∏i<j

|λi − λj |2∏

k∈[dA]

λdA−dBk 1λ∈∆dA(8.33)

where, for all integers n ≥ 1, ∆n denotes the n-dimensional simplex

∆n :=

x ∈ Rn : xi ≥ 0 for all i ∈ [n] ,∑i∈[n]

xi = 1

, (8.34)

131

and ZdA,dB is the normalization constant. We see that the joint law of the Schmidt eigenvalues ofa random pure state from the unbiased ensemble has the familiar 2D Coulomb gas form. Here,the eigenvalues are conned on the dA-dimensional simplex. In the Coulomb gas picture, weare dealing with a gas on the positive half-line with xed center of mass. e whole machinerydeveloped for invariant random matrix models (see Chapters 2 and 4) is tailored for a thoroughinvestigation of the unbiased ensemble.

Indeed, it is evident from the joint law (8.33) (see Example 2.3 and Table 7.1) that ωA isdistributed as a complex Wishart matrixWc (β = 2) with parameter c = dB/dA ≥ 1 and xedtrace trWc = 1. Using Stieltjes eorem 2.2 we easily nd that, for all dB ≥ dA ≥ 2, the mostprobable eigenvalues

λ? = arg maxλ

P(λ) (8.35)

are given by the zeroes of the following Laguerre polynomials:

L(α−1)dA

(ξx) =

dA∑ν=0

cν(−x)ν , cν =ξν

ν!

(dB − 1

dA − ν

),

α = dB − dAξ = dA(dB − 1) .

(8.36)

For more details on this nite dimensional result see [53].For large dimensional quantum systems (dA,B 1) we can exploit the results on the moments

of the Wishart ensemble and the covariance formula (7.3). Let us consider a dA-dimensionalWishart modelWc of parameter c. For this model E[trWc] ∼ cdA and Var(trWc) = O(1). Byconcentration of measure (this argument can be strenghtned), the ensemble 1

cdAWc is asymptoti-

cally distributed as ωA. For all practical purposes it is thus justied to mimic the random localpart ωA of a unitarily invariant random pure state as a rescaled Wishart random matrix.

eorem 8.8. Let |ψ〉 ∼ Unif(HA ⊗ HB). If dBdA→ c ≥ 1 for dA → ∞, then the rescaled

empirical spectral density of ωA

%dAωA(λ) =1

dA

∑i∈[dA]

δ(λ− dAλi) (8.37)

converges to the Marcenko-Pastur law with parameter c:

%dAωA(λ) c

2πλ

√(λ− λ−)(λ+ − λ) 1 λ−<λ<λ+ , with λ± =

(1±√c)

2

c, (8.38)

as dA → ∞. Moreover, the family of linear Renyi’s entropies Σq(ωA) q≥0 is jointly asymptoti-cally Gaussian with mean and covariance structure given by

E[Σq] =1

(cdA)q−1

q∑s=1

cs−1Nar(q, s) (8.39)

Cov(Σp,Σq) =1

(cdA)p+q

[Jc(p, q)−

Jc(1, p)Jc(1, q)

Jc(1, 1)

], (8.40)

132

where Nar(q, s) are the Narayana numbers (see (2.135)):

Nar(q, s) =1

s

(q

s− 1

)(q − 1

s− 1

), (8.41)

and

Jc(`,m) =(1 +

√c)`+m

Γ(`)Γ(m)

∑j=0

m∑t=0

(`

j

)(m

t

)(1−√c

1 +√c

)j+tReA`mjt , (8.42)

with A`mjt given in (7.63).

Proof. e proof of the (8.38) is an immediated consequence of the convergence of the ESdfor complex (β = 2) Wishart matrices Wc. Moreover, the family trWq

c : q ≥ 0 is jointlyasymptotically Gaussian. e strategy is then the following: consider the joint ditribution of(trWc, trWp

c , trWqc ), and then take the conditional distribution on trWc = 1. en, the average

value of the linearized Renyi’s entropies Σq(|ψ〉) is given in terms of the moments of the rescaledWishart model (see (2.134):

E[Σq] = E[trωqA] (8.43)

∼ 1

(cdA)q−1E[trWq

c ] (8.44)

∼ 1

(cdA)q−1

q∑r=1

cr−1

r

(q

r − 1

)(q − 1

r − 1

). (8.45)

Using the covariance formula (7.3) one can compute the covariances

Cov(Σp,Σq) =1

(cdA)p+q

(Cov(trWp

c , trWqc )− Cov(trWc, trWp

c )Cov(trWc, trWqc )

Var(trWc)

)=

1

(cdA)p+q

[Jc(p, q)−

Jc(1, p)Jc(1, q)

Jc(1, 1)

], (8.46)

where we have used Corollary 7.4 with λ± = (1±√c)

2 and β = 2.

For instance, the local purity πAB(|ψ〉) = Σ2(ωA) of a random pure state |ψ〉 ∈HA ⊗HB

with dB ∼ cdA (c ≥ 1) is asymptotically Gaussian with mean and variance

E[πAB(|ψ〉)] = E[Σ2(ωA)] =1 + c

c

1

dA(8.47)

Var(πAB(|ψ〉)) = Var(Σ2(ωA)) =2

c2

1

d4A

. (8.48)

We notice that (8.47) agrees with the asymptotics of (8.17). e statistical behavior of Renyi’sentropies

Sq(ωA) =1

1− qlog Σq(ωA) , q > 1 , (8.49)

is easily inferred from the behavior of Σq(ωA).

133

Corollary 8.9. Let |ψ〉 ∼ Unif(HA ⊗HB). If dBdA→ c ≥ 1 for dA → ∞, then the family of

Renyi’s entropies Sq(ωA) q≥0 is jointly asymptotically normal with

Sq ∼ AN

(1

1− qlogE[Σq],

Var(Σq)

(1− q)2E[Σq]2

). (8.50)

and covariances

Cov(Sp, Sq) =Cov(Σp,Σq)

(1− p)(1− q)E[Σp]E[Σq], (8.51)

with E[Σq] and Cov(Σp,Σq) as in eorem 8.8.

Proof. e corollary follows easily from an application of the Delta method (see for instanceSection 5.5. of [227]) on the results on the linerized Renyi’s entropies of eorem 8.8.

In Fig. 8.1 we compare the prediction (8.45)-(8.46) with numerical simulations. We concludethis section with a result about random states |ψ〉 with constrained Schmidt degeneracy m(|ψ〉)(see (8.11)). We will denote by |ψ〉 ∼ Unifm(HA ⊗HB) the unitarily invariant measure onS(HA ⊗HB) restricted to states |φ〉 with xed Schmidt degeneracy m(|φ〉) = m. ese statesare mapped by s (8.8) into the same face of the Weyl chamber.

eorem 8.10 (Random states with xed Schmidt degeneracy). Let |ψ〉 ∼ Unifm(HA⊗HB) bea random states with xed Schmidt degeneracy m(|φ〉) = m. Let us denote the k = ‖m‖0 −m0

nonzero distinct eigenvalues of |ψ〉 by ν = (ν1, . . . , νk). en the law of ν is

P(m)dA,dB

(ν) =1

Z (m)dA,dB

∏1≤i<j≤k

(νi − νj)2mimj∏`∈[k]

ν`(dB−dA+2m0)ml 1ν∈∆mk

, (8.52)

supported on

∆mk :=

x ∈ Rk : xi ≥ 0 for all i ∈ [k] ,∑i∈[k]

mixi = 1

. (8.53)

Proof. = For simplicity, let us consider P↓dA,dB

(λ) be the joint law of the ordered eigenvalues ofωA:

P↓dA,dB

(λ) =dA!

ZdA,dB

∏i<j

|λi − λj |2∏

k∈[dA]

λdA−dBk 1λ∈∆dA1 0≤λ1≤λ2≤···≤λdA . (8.54)

It is easy to recognize that the joint law (8.33) can be recovered from (8.54) by symmetrization.is choice will simplify the notation in the proof. In order to prove the theorem, it will besucient to consider the following two cases:

134

(i) P↓dA,dB

(λ |λd1 = 0): the joint law of the λi’s conditioned to have one of them equal tozero. is corresponds to the joint law of the Schmidt eigenvalues of a random pure state|ψ〉 ∼ Unifm(HA ⊗HB) be a random states with xed Schmidt degeneracy m(|φ〉) =(1, 1, . . . , 1)︸ ︷︷ ︸

dA

;

(ii) P↓dA,dB

(λ |λd1 = λd2): the joint law of the λi’s conditioned to have two coincidenteigenvalues. In this case |ψ〉 ∼ Unifm(HA ⊗HB) with Schmidt degeneracy m(|φ〉) =(0, 2, 1, 1, . . . , 1)︸ ︷︷ ︸

dA

;

Let us proceed with the standard calculation. For all X Borel subsets of RdA+ , conditional proba-bilities read:

Pr(λ1, . . . , λdA) ∈ X |λ1 = 0 = limε↓0

Pr (λ1, . . . , λdA) ∈ X | 0 ≤ λ1 ≤ ε

= limε↓0

Pr

(λ1, . . . , λdA) ∈ X ∩([0, ε] ∪ RdA−1

) Pr 0 ≤ λ1 ≤ ε

= limε↓0

´X dλP↓

dA,dB(λ1, . . . , λdA) 1λ1∈[0,ε]´

RdA dλP↓dA,dB

(λ1, . . . , λdA) 1λ1∈[0,ε]

= limε↓0

´X dλP↓

dA,dB(λ1, . . . , λdA)δε(λ1)´

RdA dλP↓dA,dB

(λ1, . . . , λdA)δε(λ1),

(8.55)

where we have used the the l’Hopital-Bernoulli rule in the last line. By apply (dB − dA) furthertimes the l’Hopital-Bernoulli rule we obtain

P↓dA,dB

(λ |λ1 = 0

)∝

∏1≤i<j≤dA−1

|λi − λj |2∏

1≤l≤dA−1

λdB−dA+2l 1λ∈∆dA

1 0=λ1≤···≤λdA .

If we write dB − dA + 2 = (dB + 1)− (dA − 1) and adjust constants we nally obtain:

P↓dA,dB

(λ |λ1 = 0) = P↓dA−1,dB+1 (λ1, . . . , λdA−1) 1 λ1=0 . (8.56)

Equation (8.56) can be readily generalized to the seing in which more than one eigenvalue arezero. e hierarchy of conditioned law of the spectrum ofωA with the condition rankωA = dA−k,for 0 ≤ k ≤ dA − 1, is:

P↓dA,dB

(λ |λ1 = · · · = λk = 0) = P↓dA−k,dB+k (λ1, . . . , λdA−k) 1 λ1=···=λk=0 . (8.57)

135

With a similar approach one can verify that if we impose that two eigenvalues, say λ1 and λ2

coincide, the two corresponding charges merge and form a single movable object with doublecharge. In term of conditioned probability law:

P↓dA,dB

(λ |λ1 = λ2) ∝∏

1≤i<j≤dA−2

|λi − λj |2∏

k∈[dA−2]

λkdA−dB

×∏

k∈[dA−2]

|λk − λ1|4 λ2(dA−dB)1 1λ∈∆dA

1 0≤λ1=λ2≤···≤λdA .

(8.58)Now it is not dicult to summarize the above results in order to obtain the claim (8.52) of thetheorem.

Remark 8.1. e presented probability densities agree with the physical intuition expected fromthe Coulomb gas analogy. Let us translate (8.56) in the language of the Coulomb gas picture. Ifwe constrain one of the dA charges to sit at the origin, the system will loose a charge (a degree offreedom) that now has to be considered as a contribute of the external eld. e remaining dA− 1charges will continue to interact via logarithmic potential. e generalization (8.57) admits ananalogue interpretation. In the same way, if λ1 and λ2 coincide, the two corresponding chargesmerge and form a single movable object with double charge. More constraints correspond tomultiple merging of the charges giving rise to (8.52), namely

P(m)dA,dB

(ν) =1

Z (m)dA,dB

e−[∑i6=j mimj log|νi−νj |+(dB−dA+2m0)

∑`∈[k] log ν`] 1ν∈∆mk

. (8.59)

As a byproduct, this discussion suggests a mechanism that enhances the level repulsion. eprojection of a unitarily invariant self-adjoint ensemble to the set of matrices with non-simplespectra “increases” the Dyson index β.

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: simulations Hn = 50L

: theory

: simulations Hn = 150L

0 5 10 15 20 251

100

104

106

108

q

nq-1E

@SqD

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: simulations Hn = 50L

: theory

: simulations Hn = 150L

0 5 10 15 20 251

104

108

1012

1016

1020

q

n2qV

arHS

qL

Figure 8.1: Comparison of the theoretical prediction and simulations for the rescaled average E[Σq(ωA)] and varianceVar(Σq(ωA)) of the linearized Renyi’s entropies Σq(ωA) = trωqA of a random pure state |ψ〉 ∈ HA ⊗HB . Heredim HA = n and dim HB = cn with n = 50 (blue diamonds) and n = 150 (red circles) both at c = 2. Using theDumitriu-Edelman tridiagonal trick [71] we have sampled N = 104 spectra of reduced states ωA (the tridiagonalprocedure has been suitable modied by including an overall normalization by the trace). e theory refers to Equations(8.45) and (8.46).

Chapter 9

Polarized Ensembles: typical purity

Random quantum states has gradually become an intensively studied subject. More precisely,people are interested in the typical properties exhibited by some particular ensembles of randomstates. e setup of the problem is specied by the choice of the ensemble, namely the speciclaw of the random states. Once the setup is established, several tailored techniques can be appliedin order to explore the typical properties of the ensemble. e most intensively studied setupis the unbiased ensemble (see Denition 8.5). Such an expression refers to random pure states|ψ〉 ∈ S(H ) distributed according to the unitarily invariant measure on the pure state space.In the previous Chapter we have seen how the typicality for the unbiased ensemble can beinvestigated by exploiting the large technology provided by RMT .

A natural question is whether this ensemble of pure random states can be used to constructdierent, more complicated, ones. Various approaches have been proposed independently byseveral groups that have introduced dierent physically motivated measures on the space ofpure states. Recently, Zyczkowski et al. [241] have analyzed some kind of structured ensembles ofrandom pure states on composite systems that are invariant under local unitary transformations.From another perspective, De Pasquale et al. [63] have proposed a classical statistical mechanicsapproach in order to explore the isopurity manifolds of random states. In the same spirit, Muelleret al. [161] have recently investigated ensembles of random pure states with xed expectation valueof some observable, in the framework of the concentration of measure phenomenon. However,there are still many obstructions in carrying out these programs, and the links among them is notyet clear.

Recently we have proposed a general class of polarized ensemble of random states [54] asa step toward new scenarios beyond the unbiased ensemble. is step was motivated as anoperational way to capture the isopurity manifolds, and turns out to be in particular cases similarto the structured ensembles proposed in [241]. e idea is to exploit a genuine operation at handin the Hilbert space, namely superposition of vector states. A natural program is to take the sum(as an operation in the Hilbert space H ) and to push it forward as a composition law in the spaceof probability measuresM(S(H ))) on the sphere of pure states.

is Chapter is organized as follows. Firstly, we introduce the general scheme, the basic ideas

137

138

and the main questions which one is interested in (Section 9.1). en we introduce the concept ofpolarized ensembles and we characterize them by using the local purity of a subsystem (Section9.2). In particular we study the polarization of the unbiased ensemble by a tunable randomperturbation. e most relevant cases are the separable and the maximally entangled perturbation.is approach will be used both for the study of the ensembles and for the description of anecient procedure for sampling typical states from an isopurity manifold (and, therefore, witha xed value of bipartite entanglement). In the next Chapter we will investigate the spectralproperties of these polarized ensembles.

9.1 Polarized ensembles of pure states

Consider a bipartite quantum system described in a nite-dimensional Hilbert space H with atensor product structure

H = HA ⊗HB. (9.1)

We will consider, without loss of generality, the case with

dim HA = dA ≤ dim HB = dB, (9.2)

whence dim H ≡ d = dAdB .For the purposes of the following discussion, we briey recall the general procedure to dene

an ensemble of random pure states. For any pure state |ψ0〉 ∈H we can consider the orbit ofa group of transformations G through |ψ0〉: G|ψ0〉 ≡ g |ψ0〉g∈G. We will consider compactgroups acting on the compact manifold S(H ). is means that the Haar measure on such agroup G induces in a natural way a G-invariant probability law on the orbit G|ψ0〉. A G-invariantrandom state |ψ〉 ∼ Unif(G|ψ0〉) is thus dened. We stress the fact that the orbit G|ψ0〉 and, byconsequence, the random state |ψ〉 ∼ Unif(G|ψ0〉) depends in general on the choice of the ducialvector |ψ0〉.

ere are two privileged groups that act on a bipartite Hilbert space H : the full unitary groupU(H ) and the group of local unitaries U(HA)×U(HB). e unitary group U(H ) correspondsto the group of global transformation on S(H ), while the local unitaries correspond to localtransformations on HA and HB separately. e orbit of the full unitary group through any stateis the whole states space. e Haar measure µd on the unitary group U(H ) induces a unitarilyinvariant (unbiased) law on pure states supported on S(H ). We already denoted such a randomstate as |ψ〉 ∼ Unif(H ).

On the other hand, the space of pure states |ψ〉 ∈ S(HA ⊗HB) is foliated in submani-folds labelled by the multi-index m(|ψ〉) (see (8.11)) encoding the degeneracy of the Schmidtcoecients [200, 24]. If we denote by µdA and µdB the Haar measure on U(HA) and U(HB) re-spectively, the product measure µdA×µdB is two-side invariant under the action of local unitariesUA ⊗ UB ∈ U(HA) × U(HB). is measure naturally induces a U(HA) × U(HB)-invariantprobability law on each orbit of local unitaries (see the discussion in Section 8.1 and eorem8.3). We will use the notation |ψ〉 ∼ Unif(U(HA) × U(HB)|ψ0〉). Two somehow privilegedorbits of local unitaries are the manifold of separables states OSep and the manifold of maximally

139

entangled statesOME. ese correspond to the extremal situations the degeneracy of the Schmidtcoecients s(|ψ〉) ism = (dA−1, 1) andm = (0, dA), respectively (see eorem 8.3). Moreoverfor these particular choices an equal spectrum of the reduced density matrices is equivalent toequal value of entanglement measures (the orbit of local unitaries coincide with the isoentangledmanifold).

Let us focus on the general situation in which the pure state of the quantum system has theform

|ψ〉 = α |ψ1〉+ β |ψ2〉 , (9.3)where |ψ1〉 , |ψ2〉 ∈H are random states, sampled according to arbitrary probability measures.Once the probability distributions of |ψ1〉 and |ψ2〉 are specied, the random variable |ψ〉 ischaracterized by a well dened distribution.

In general, depending on the ensembles chosen for sampling |ψ1〉 and |ψ2〉, the state |ψ〉dened in (9.3) will exhibit very dierent properties. Let us briey outline the relevant features insome cases of particular interest.

If both |ψ1〉 and |ψ2〉 are sampled according to the unbiased ensemble law(|ψi〉) = Unif(H )(i = 1, 2), then |ψ〉 is also a random state whose distribution is invariant under the action of theunitary group U(H ) (independently on the values of α and β).

e opposite situation occurs when |ψ1〉 is a xed pure state and |ψ2〉 ∼ Unif(H ). In thiscase the weights α and β are relevant; if |α| |β| the unbiased ensemble becomes “polarized”along the direction dened by |ψ1〉. is polarization phenomenon is of particular interest if onewants to study the deviation of the properties of an ensemble of quantum states from a xedreference state. An intermediate situation is realized when |ψ1〉 ∼ Unif(U(HA)× U(HB)|ψ0〉)for some ducial vector |ψ0〉 ∈H and |ψ2〉 ∼ Unif(H ). As explained above, this ensemble isdened by the specic orbit of U(HA)× U(HB) through a ducial vector |ψ0〉. We will choosethe interesting cases in which |ψ0〉 is separable (i.e. |ψ1〉 ∼ Unif(OSep)) or maximally entangled(i.e. |ψ1〉 ∼ Unif(OME)).

9.2 One-parameter ensembles

Let us turn to the general scheme that we want to investigate. Let us consider the superpositionof two independent random states (9.3). In the following it will be clear that, as long as we studythe entanglement spectrum, the relative phase of |ψ1〉 and |ψ2〉 is not relevant, and thereforewe can limit ourselves to the case of α and β both real. For such a reason we will focus on thefollowing random superposition

|ψ〉 =1√C

[ε |ψ1〉+

√1− ε2 |ψ2〉

]∈H , (9.4)

where |ψ1〉 and |ψ2〉 are independent random states whose laws we do not yet specify, ε ∈ [0, 1]is a tunable parameter and C = 1 + 2ε

√1− ε2 Re 〈ψ1 |ψ2〉. For this ensemble we are going to

study the local purity of one of the parties

πAB(|ψ〉) = trω2A , (9.5)

140

where ωA = trB |ψ〉 〈ψ| is the reduced density matrix on HA. e upper bound πAB = 1 andthe lower bound πAB = 1/dA correspond, respectively, to separable and maximally entangledstates.

We preliminary observe that, if one of the two vectors, say |ψ2〉 is uniformly distributed onthe sphere of state, i.e. |ψ2〉 = Unif(H ), then C = 1 with overwhelming probability. Indeed,since law(|ψ2〉) = law(− |ψ2〉) the overlap of the two states vanishes on average,

E [〈ψ1 |ψ2〉] = 0. (9.6)

Moreover the map |ψ2〉 7→ 〈ψ1, ψ2〉 has Lipschitz constant η = ‖ψ1‖. en, by Levy’s lemma 8.5:

Pr (|C − 1| ≥ δ) = Pr(

2ε√

1− ε2|Re 〈ψ1 |ψ2〉| ≥ δ)

≤ Pr

(|〈ψ1 |ψ2〉| ≥

δ

2ε√

1− ε2

)≤ 2 exp

(− 2dAdBδ

2

36ε2(1− ε2)π3

). (9.7)

9.3 Typical local purity

In this section we will focus on the consequences of the polarization of the ensemble on theproperties of bipartite entanglement between subsystems HA and HB .

9.3.1 Local purity of one-parameter polarized ensembles

We are interested in the expectation value of the purity πAB(ψ) of

ψ = ε |ψ1〉+√

1− ε2 |ψ2〉 (9.8)

given a bias 0 ≤ ε ≤ 1. Due to concentration of measures, for large dA, dB this quantity willbe the typical purity of the polarized ensemble (9.3). We emphasize that, since we are focusingon the typical features of an ensemble of random pure states, any statement in the paper hasto be considered in the large size limit, dA, dB →∞. In this limit, the ensemble of vectors (9.8)is an ensemble of physical states, in the sense that it consists of unit vectors ‖ψ‖2 = 1 withoverwhelming probability (see (9.7)).

Equivalently, the positive operator

ψψ† = ε2 |ψ1〉 〈ψ1|+ (1− ε2) |ψ2〉 〈ψ2|+ ε

√1− ε2 (|ψ1〉 〈ψ2|+ |ψ2〉 〈ψ1|) . (9.9)

is a projection operator with overwhelming probability. We will use the following notation:

ωA = trB ψψ† = ε2ω1 + (1− ε2)ω2 + ε

√1− ε2S,

ω1 = trB |ψ1〉 〈ψ1| ,ω2 = trB |ψ2〉 〈ψ2| ,S = trB (|ψ1〉 〈ψ2|+ |ψ2〉 〈ψ1|) . (9.10)

141

By tracing over HB and performing a straightforward calculation, we obtain the purity

πAB(ψ) = trA ω2A

= ε4trω21 +

(1− ε2

)2trω2

2 + ε2(1− ε2

)trS + 2ε2

(1− ε2

)tr(ω1ω2

)+2ε3

√1− ε2tr

(ω1S

)+ 2ε

(1− ε2

)3/2tr(ω2S

). (9.11)

For the superposition (9.8) we have the following result:

eorem 9.1. Let |ψ1〉 and |ψ2〉 independent random pure states. Suppose that |ψ2〉 is symmetri-cally distributed, law(|ψ2〉) = law(− |ψ2〉), and

E |ψ2〉 〈ψ2| =1

dAdB. (9.12)

en, the average purity of (9.8) is

E[πAB(ψ)] = ε4π1 +(1− ε2

)2π2 + 2ε2

(1− ε2

)π0 , (9.13)

where we have denoted πi = E[πAB(|ψi〉)] (i = 1, 2) and the geometric factor π0 (independent of|ψ1〉, |ψ2〉 and ε) is

π0 =1

dA+

1

dB(9.14)

Proof. A direct calculation shows that the only non vanishing terms in the expectation valueof (9.11) are

E[trω21] = π1, (9.15)

E[trω22] = π2, (9.16)

E[tr (ω1ω2)] =1

dA, (9.17)

E[trS2] =2

dB. (9.18)

Indeed the other terms are zero since E[|ψ2〉] = E[− |ψ2〉]. By plugging (9.15)-(9.18) into (9.11)we nally get the value of the purity πAB(|ψ〉) (9.13). Let us derive in details (9.17)-(9.18). Letthe dAdB complex components of |ψ1〉 and |ψ2〉 (in a given basis) be xiµ and yiµ, respectively,where i ∈ [dA] and µ ∈ [dB] (hereaer latin indices will run in [dA] while greek indices will varyin [dB]). e hypotheses on the law of |ψ2〉 imply

E [yiµ] = 0 , E [yiµyjν ] = 0 , and E[yiµy

∗jν

]=

1

dAdBδijδµν , (9.19)

for all i, j, µ, ν. Let us consider now the terms (9.17)-(9.18). e reduced states ω1 = trB |ψ1〉 〈ψ1|has components

∑µ xiµx

∗jµ (i, j ∈ [dA]), while ω2 = trB |ψ2〉 〈ψ2| has components

∑µ yiµy

∗jµ

142

(i, j ∈ [dA]). en

E[tr (ω1ω2)] = E[∑i,j,µ,ν

xiµx∗jµyjνy

∗iν ]

=∑i,j,µ,ν

E[xiµx∗jµ]E[yjνy

∗iν ]

=1

dAdB

∑i,j,µ,ν

E[xiµx∗jµ]δijδµν

=1

dAdB

∑i,µ,ν

E[|xiµ|2] =1

dAdB

∑ν

1 =1

dA. (9.20)

e last termS = trB (|ψ1〉 〈ψ2|+ |ψ2〉 〈ψ1|) is a random matrix with entries∑

µ xiµy∗jµ + yiµx

∗jµ.

By squaring it and taking the trace we obtain

E[trS2] = E[∑i,j,µ,ν

(xiµy

∗jµ + yiµx

∗jµ

)(xjνy

∗iν + yjνx

∗iν)]

= 2∑i,j,µ

E[|xiµ|2]E[|yjµ|2]

=2

dAdB

∑i,j,µ

E[|xiµ|2] =1

dAdB

∑j

1 =2

dB. (9.21)

Adding up all the pieces we obtain the result (9.22).

eorem 9.1 generalizes the result in [54] that we quote here:

Corollary 9.2. Let |ψ1〉 ∼ µ(1) a random pure state and |ψ2〉 ∼ Unif(HA ⊗HB) a uniformlydistributed state independent of |ψ1〉. e average purity of the sum (9.8) is

E[πAB(|ψ〉)] = ε4π1 +(1− ε4

)π0 +O

(1

dAdB

), (9.22)

where we have denoted π1 ≡ E[πAB(|ψ1〉)] .

Proof. Clearly |ψ2〉 satises the hypotheses of eorem 9.1. It remain to computeπ2 = E[πAB(|ψ2〉)]and show that

π2 = π0 +O(

1

dAdB

). (9.23)

We notice that for the computation with a uniform measure on the sphere we need the four-pointcorrelation. Indeed for the unit vector |φ〉 uniformly distributed on the unit sphere the fourthmoment reads

E[yiµy∗jµyjνy

∗iν ] =

1

dAdB(dAdB + 1)(δij + δµν) , (9.24)

143

Using this result the explicit computation of π2 read as follows

E[trω22] = E[

∑i,j,µ,ν

yiµy∗jµyjνy

∗iν ]

=1

dAdB(dAdB + 1)

∑i,j,µ,ν

(δij + δµν)

=dA + dBdAdB + 1

. (9.25)

Let us consider for instance a vector obtained as a superposition (9.3) where, in particular, axed pure state is superposed to an unbiased random one. We will get the following one-parameterensemble

ψ =[ε1 +

√1− ε2UAB

]|ψ1〉 (9.26)

= ε |ψ1〉+√

1− ε2 |ψ2〉 (9.27)

where |ψ1〉 ∈HA ⊗HB is xed ducial vector, ε ∈ [0, 1] is a tunable parameter, 1 is the identityoperator on HA ⊗HB , and UAB ∈ U (HA ⊗HB) is a random unitary, sampled according tothe Haar measure on the full unitary group. e state

|ψ2〉 = UAB |ψ1〉 ∼ Unif(H ) (9.28)

is therefore a random state distributed according to the unitarily invariant measure on pure states.We do not care about the exact normalization of ψ since we have already proved that ψ is a unitvector with overwhelming probability (see (9.7)). Notice that for ε = 0 one recovers the unbiasedensemble. On the other hand, values of ε > 0 play the role of an oset which parametrizes, asdiscussed in the previous section, the degree of polarization of the ensemble in the direction of|ψ1〉. e local purity of ψ is given by Corollary 9.2.Remark 9.1. Even if |ψ1〉 is substituted by a state belonging to its local orbit |ψ′1〉 = UA⊗UB |ψ1〉,with (UA, UB) ∈ U(HA)× U(HB), the value of the average local purity given by (9.22) is notaected. In other words, the one-parameter ensemble of random states

ψ =[εUA ⊗ UB +

√1− ε2UAB

]|ψ1〉 , (9.29)

where UA ⊗ UB is a random local unitary, has average purity given by formula (9.22), withπ1 = πAB(|ψ1〉).

Having in mind this remark, we can establish the following central result.

144

eorem 9.3. Let |φ0〉 ∈ HA ⊗HB with Schmidt coecients λ1, . . . , λdA . Consider UAB andUA ⊗ UB be independent random unitary operators distributed according to the Haar measure onU(HA ⊗HB) and U(HA)× U(HB), respectively, and let

ψ =[εUA ⊗ UB +

√1− ε2UAB

]|φ0〉 (9.30)

= ε |ψ1〉+√

1− ε2 |ψ2〉 . (9.31)

en

E[πAB(ψ)] = ε4∑i∈[dA]

λ2i +

(1− ε4

)π0 +O

(1

dAdB

). (9.32)

In particular, if |φ0〉 is separable or maximally entangled, then |ψ1〉 ∼ Unif(OSep) or |ψ1〉 ∼Unif(OME) respectively, and

E[πAB(ψ)] =

π0 + ε4 (1− π0) , if |φ0〉 is separableπ0 − ε4

dB, if |φ0〉 is maximally entangled .

(9.33)

Moreover the average value of πAB(ψ) is also its typical value:

Pr ∣∣∣πAB(ψ)−E[πAB(ψ)]

∣∣∣ > δ≤ 2 exp

(− dAdBδ

2

32(1− ε2)

). (9.34)

Proof. e average values (9.32) and (9.33) follows from the thesis of Corollary 9.2. In order toprove the typicality we observe that, for all |ψ1〉 ∈ S(H ), the function f : S(H ) 3 |ψ2〉 7→πAB(ε |ψ1〉+

√1− ε2 |ψ2〉) has Lipschitz constant η ≤ 4

√1− ε2. en, using Levy’s Lemma 8.5

Pr ∣∣∣πAB(ψ)−E[πAB(ψ)]

∣∣∣ > δ∣∣∣ |ψ1〉

≤ 2 exp

(− dAdBδ

2

32(1− ε2)

). (9.35)

e concentration result (9.34) descends from the trivial observation

Pr ∣∣∣πAB(ψ)−E[πAB(ψ)]

∣∣∣ > δ= E|ψ1〉

[Pr ∣∣∣πAB(ψ)−E[πAB(ψ)]

∣∣∣ > δ∣∣∣ |ψ1〉

].

9.3.2 Generation of random pure states with xed local purity

According to eorem 9.3 we shall consider the one-parameter ensemble polarized in the “direction”of the separable or maximally submanifolds of pure states. We recall here that for the unbiasedensemble (no polarization, ε = 0) the typical purity is

E[πAB(ψ)] = π0 +O(

1

dAdB

). (9.36)

145

Let us focus rstly on the ensemble polarized with a random separable states

ψ =[εUA ⊗ UB +

√1− ε2UAB

]|φSep〉 , (9.37)

where UA and UB are random local unitaries and UAB is a random global unitary transformation(statistically independent). e typical purity of the one-parameter ensemble with a separablepolarization (10.22) reads

E [πAB(ψ)] = π0 + ε4 (1− π0) . (9.38)e other extreme case is given by a maximally entangled perturbation |φME〉. en we aredealing with a polarized ensemble of the form

|ψ〉 =[εUA ⊗ UB +

√1− ε2UAB

]|φME〉 . (9.39)

Such a polarization decreases the typical purity of the unbiased ensemble to the value

E [πAB(ψ)] = π0 −ε4

dB. (9.40)

ese results suggest a very inexpensive strategy for generating random pure states with xedvalue of the purity πAB by suitably polarized an unbiased pure state with a polarization uniformlydistributed on OSep or OME. Indeed, the numerical sampling of pure states |ψ〉 parametrized bya xed value of πAB will proceed through the steps outlined in Experiment 4. is strategy yields

Experiment 4 Sampling of random pure states |ψ〉 with a xed purity πAB1: |ψ2〉 ∼ Unif(HA ⊗HB);2: Set π0 = 1

dA+ 1

dB;

3: if πAB ≥ π0

4: π1 = 1 and |ψ1〉 ∼ Unif(OSep);5: else if πAB ≤ π0

6: π1 = 1/dA and |ψ1〉 ∼ Unif(OME);7: end if8: Choose ε ∈ [0, 1] such that πAB = ε4π1 +

(1− ε4

)π0;

9: Polarized ensemble: ψ = ε |ψ1〉+√

1− ε2 |ψ2〉;10: Normalize ψ → |ψ〉.

an ecient and simple sampling of random pure states with xed value of purity, and paves theway to further explorations and a deeper characterization of the geometry of isopurity manifolds.

In gure 9.1 (upper panel) the analytical formulas (9.38) and (9.40) are compared to the MonteCarlo results for the values of πAB obtained by sampling pure states through the procedureoutlined above. e comparison shows clearly the eciency of the sampling procedure inproviding the correct behavior of the typical purity vs the bias ε with quite small uctuationsalready for dimensions dA = dB = 30. Such uctuations around the average are more evidentfor smaller systems. See gure 9.1 (lower panel) for the case dA = dB = 8.

Maximally EntangledΦ ent >

SeparableΦ sep >

1 0.5 0 0.5 10.0

0.2

0.4

0.6

0.8

1.0

Ε4

ΠA

B

N=30 M=30

Maximally EntangledΦ ent >

SeparableΦ sep >

1 0.5 0 0.5 10.0

0.2

0.4

0.6

0.8

1.0

Ε4

ΠA

B

N=8 M=8

Figure 9.1: Typical purity of polarized ensembles vs ε4, depending on the bias |φ0〉. We compare the analyticalprediction (continuous lines) with the numerical values of πAB (sample mean and error bars) obtained from thesampling procedure described in the text. We considered balanced bipartitions with size dA = dB = 30 (top)and dA = dB = 8 (boom). In both cases the number of realizations used to perform the ensemble average isn = 104. Right side: the continuous line represents the analytical prediction for the purity of an ensemble polarizedby a separable state (9.38). Depending on the value of the parameter ε the purity ranges from the unbiased valueπ0 = d−1

A + d−1B to the maximum 1. Le side: the continuous line represents the analytical prediction for the purity

of an ensemble polarized by a maximally entangled state (9.40). As ε increases the typical value of the purity decreasesfrom π0 to the minimum 1/dA. e error bars represent the standard deviations of the numerical simulations from theestimated average. Such uctuations are exponentially suppressed as the dimensions dA, dB increase, according to(9.34).

Chapter 10

Polarized Ensembles: spectralproperties

So far we have studied the polarized ensemble in terms of the local purity. We will now considersome more detailed spectral properties. We warn the reader that we are far from having a completedescription of the spectral density of the reduced state ωA for the polarized ensembles. Technically,this is essentially due to the lack of knowledge of simple expressions for the spectral distributionfor non-central Wishart random matrices [162]. However, many techniques can be borrowed bythe large dimensional spectral analysis [12], to prove some numerical evidences and to predictphase transitions in the spectral density of large dimensional ensembles. To this regard it isworth stressing that, since polarized ensembles have a concrete matrix representation, these phasetransitions can be explicitly checked and veried numerically. In this sense this chapter is largelyphenomenological. We will investigate some “Facts”, namely some numerical evidences, and wewill try to provide a mathematical description of them. We will consider here some instances ofpolarized ensemble in the large size limit dA, dB →∞ with dB/dA → c ≥ 1 xed and nite:

Polarization of the unbiased ensemble with a maximally entangled state: in Section 10.1we will investigate in more details the density of states of this ensemble. By characterizingits Stieltjes transform and its support, we will demonstrate a phase transition in the controlparameter ε.

Polarization of the unbiased ensemble with a separable state: in Section 10.2 we will provethe existence of a phase transition, namely the existence of an “outlier” in the spectrum forsuciently large values ε of the polarization.

Superposition of random maximally entangled states: we will derive in Section 10.3 thedensity of states of the ensemble. Remarkably we will see that the limiting density isindependent of the dimension dB .

147

148

10.1 Adding noise to a maximally entangled state

We study the coherent superposition (9.27), where |ψ1〉 ∼ Unif(OME) is a random maximallyentangled state and |ψ2〉 ∼ Unif(HA ⊗HB) is uniformly distributed. It is convenient to writesuch a superposition in the following notation

ψ =[εUA ⊗ UB +

√1− ε2UAB

]|φME〉 , (10.1)

where the ducial state |φME〉 is such that

trB |φME〉 〈φME| =1A

dA, (10.2)

UA and UB are random local unitaries and UAB is a random global unitary transformation(UA, UB and UAB are jointly independent). Values of ε > 0 play the role of a bias which polarizesthe unbiased ensemble in the “direction” of maximally entangled states. In the large size limit, thepurity of the polarized ensemble (10.1) concentrates around its mean value (9.40).

We now focus on the spectral properties of dAωA = dAtrB ψψ†, namely the empirical spectral

density%dAωA(λ) =

1

dA

∑i∈[dA]

δ(λ− dAλi) , (10.3)

where λi, i ∈ [dA] are the eigenvalues of ωA. Here we summarize the two extremal situations:

Unbiased ensemble For ε = 0 we are considering the unbiased ensemble. en, according toeorem 8.8, we have

%dAωA(λ) c

2πλ

√(λ− λ−)(λ+ − λ) 1 λ−<λ<λ+ , with λ± =

(1±√c)

2

c. (10.4)

Maximally entangled ensemble For ε = 1 we are considering a random maximally entangledstate. en

%dAωA(λ) = δ(λ− 1) , for all dA, dB, c . (10.5)

e following evidence summarizes the behavior of %dAωA for generic values of ε ∈ (0, 1).

Fact 10.1 (A numerical evidence). For balanced bipartitions (dA = dB), for large dA, the ESd %dAof dAωA = dAtrB ψψ

† (for the ensemble (10.1)) is compactly supported on the positive half-line.For large values of ε > 0 this ESd is supported on supp % = [x−(ε), x+(ε)] with x−(ε) > 0.However there exists a critical value ε? > 0 such that below this threshold ε < ε? we havex−(ε) = 0. e numerical experiments suggests that ε? ' 1/

√2.

A common tool to analyze the ESd %dA(λ) is provided by its Stieltjes transform sdA(z) (seeSection 2.2). e success of the Stieltjes transform is due to its nice analytical properties and to aclassical continuity theorem relating the convergence of a sequences of ESd’s to the convergence

149

of the sequence of their Stieltjes transforms. e drawback is that, apart from a few exceptionalcases, the Stieltjes inversion formula (2.17) that allows to recover the limit ESd from the limitStieltjes transform is hard to work out. However, in the large dimensional limit, the study ofthe monotonicity of the Stieltjes transform sdA(z) gives informations about the support of thelimiting spectral distribution (a method known as qualitative analysis) [154, 12]. A result withsuch a avour is the following control on the support of the spectral measure of the reduceddensity matrix.

eorem 10.1. Let us consider the ensemble (10.1) and the rescaled ESd of ωA (10.3). For dA = dB ,the limiting spectral measure % = limdA %dA is supported on the interval [x−(ε), x+(ε)] of thepositive half-line, whose edges are given by

x±(ε) = max

0,

[11

4− 1

8ε2− 13ε2

8±√

1 + 6ε2 − 7ε4

8

(7 +

1

ε2

)] . (10.6)

à à à à à à à à à à à à à à à à à à à à à à à à àà

à

à

à

à

à

à à à à à à à à à à àà

àà

àà

à

à

à

à

à

à

à

à

à

à

à

à

à

à

à

á á á á á á á á á á á á á á á á á á á á á á á á á áá

á

á

á

á

á á á á á á á á á á á á á á á áá

áá

áá

áá

á

á

á

á

á

á

á

á

Ε=1 2

à : dim HA = 500

á : dim HA = 50

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

Ε

x ±HΕL

Figure 10.1: For the ensemble (10.1) the edges x±(ε) of the support of the limiting spectral distribution are givenby Eq. (10.6). For equally weighted superpositions (ε? = 1/

√2) the lower edge vanishes, x−(1/

√2) = 0. At this

critical value the typical purity is πAB = 7/4N . e solid lines are the analytical results, Eq. (10.6), predicted by thequalitative analysis of the Stieltjes transform. We have compared the analytical prediction with numerical simulations.e lled and empty markers refer respectively to the case dimHA = 500 and dimHA = 50. e points show thatin the large dimensional limit the support of the empirical spectral measure approaches the limiting support delimitedby x±(ε). Each point is a sample mean (with the sample standard deviation bar) over N = 103 random states.

Remark 10.1. ese ensembles exhibit a threshold behavior in the tunable parameter. Whenε ≤ ε? (balanced superposition of the unbiased ensemble and a maximally entangled states) the

150

lower edge of %dA becomes zero x− = 0 (while x+ = 27/8) for large dA. is critical value isε? = 1/

√2 as suggested by numerical evidence (see Fact 10.1). By further lowering ε, that is,

by increasing the unbiased contribution√

1− ε2UAB |φME〉, the positive matrix ωA becomesmore and more similar to the typical reduced state of the unbiased ensemble. e balancedsuperposition, with equal weights ε = 1/

√2, corresponds by (9.40) to a typical local purity

E [πAB(ψ)] = 74dA

(ε = 1/√

2) and separates two dierent regimes, as shown in Figure 10.1.We notice that the lower edge x−(ε) is not analytic in ε. Indeed the third derivative of x−(ε) isdiscontinuous at ε?:

0 =dx−dε

∣∣∣∣ε↑ε?6= dx−

dε

∣∣∣∣ε↓ε?

=219/2

32. (10.7)

In order to prove the eorem we invoke a general result on the Information-Plus-Noise-Typematrices (eorem 10.2) due to Dozier and Silverstein [67]. Lemma 10.3 will clarify the connectionbetween our ensembles and Information-Plus-Noise matrices.

eorem 10.2 (Information-Plus-Noise Type Matrices [67]). Assume that for all n ≥ 1:

a) Xn is a n×m random matrix, whose entries Xiν are i.i.d. with E[|Xiν −EXiν |2] = 1;

b) Rn is a n×m is a random matrix, independent of Xn;

c) e ESd of m−1RnR†n weakly converge %

m−1RnR†n µ, a.s., to a non random probability

distribution µ ∈M(R+).

c) m = m(n) and the limit limn→∞mn = c > 0 exists.

d) Consider the Information-Plus-Noise n× n random matrix

Hn =1

m(Rn + σXn) (Rn + σXn)†

where σ > 0.

en the ESd of Hn converges %Hn %, a.s., to a non random probability distribution % whoseStieltjes transform s = s(z) satises the integral equation

s =

ˆdµ(t)

(1 + σ2cs)−1t− (1 + σ2cs) z + σ2 (1− c), (10.8)

for any z ∈ C with Im z > 0.

Explicit solutions for the EDS by the inversion formula (2.17) can be derived in a few cases.Oen one is limited to deal with some qualitative features of the limiting EDS. A systematic wayof determining the support of the limiting distribution is provided by the so-called qualitativeanalysis (see [154]).

Indeed, suppose that sν(z) (for z /∈ suppµ) is the Stieltjes transform of a probability measureν ∈ M(R). e crucial observation is the following: on the segment of the real axis comple-mentary to supp ν the function sν(x) =

´dν(t) (x− t)−1 (x ∈ Rr supp ν) is real, continuous

151

and nonincreasing. Hence, a systematic way of determining the support of ν is the following.Write the inverse z(s) of sν(z). is inverse is dened also on those segments of the real axisoutside supp ν. On those segments we will denote the inverse as x(s). Obviously, x(s) is alsoreal, continuos and monotonically decreasing. Furthermore, its range of values is the complementof supp ν. us we have the following recipe in order to determine supp ν:

(i) Find the inverse x(s) of the Stieltjes transform (dened on the real range of sν );

(ii) plot x(s);

(iii) remove all intervals on the vertical axis where x(s) is decreasing;

(iv) what is le is the support of ν.

Let us see how the qualitative analysis works in our situation. Here we somehow know theStieltjes trasform s(z) of the limiting EDS % of Hn. In order to simplify the computation weintroduce the change of variable b(z) = 1 + σ2cs(z). In this new variable (10.8) reads

b− 1

σ2c= b

ˆdµ(t)

t− b2z + bσ2 (1− c). (10.9)

erefore we may express the inverse x(s) of s in terms of b:

x(b) =1

b2s−1µ

(1

σ2c

(1− 1

b

))+

1

bσ2 (1− c) . (10.10)

In the special case (the one useful for our scope) µ(t) = δ (t− 1) and then sµ(z) = (1− z)−1,and then (10.10) simplify

x(b) =1

b2(b− 1)

(b(1− σ2c

)− 1)

+1

bσ2(1− c) (10.11)

Once such an inverse is obtained we can perform the qualitative analysis. Note that the anechange of variable b = 1 + σ2cs does not aect the steps of the qualitative analysis. ereforethe extremal points of x(b) give the desired information about the support of %.Remark 10.2. Hypothesis a) of eorem 10.2 is clearly satied by the standard Gaussian lawXiν ∼ NC(0, 1). Here we recall that a vector φ ∈ HA ⊗ HB with indendent componentsφiν ∼ NC(0, 1/dAdB) asymptotically concentrates on the unit sphere of HA ⊗HB (for moredetails on this Gaussian approximation see Section 8.2 and the estimate (8.28)).

Lemma 10.3. Let us consider

ψ =[εUA ⊗ UB +

√1− ε2UAB

]|φME〉 , (10.12)

with the same notation as before. en, in the large size limit dA, dB → ∞, with dB/dA → c, thelimiting Stieltjes transform of dAωA is the solution of

s =1 +

(1− ε2

)cs

ε2 − (1 + (1− ε2) cs)2 z + (1− ε2) (1 + (1− ε2) cs) (1− c)(10.13)

152

Proof. e statement is a consequence of the above theorem and the rescaling property of Stieltjestransform:

sαH(z) =1

αsH(z/α) , for all diagonalizable matrices H and α ∈ C . (10.14)

e case ε = 0 is trivial (ωA = 1/dA for all dA). For ε > 0 we use the Gaussian approximationand consider the following ensemble

ψ = εφ+√

1− ε2 |φME〉

= ε

[φ+

√1− ε2ε

|φME〉

], (10.15)

with φ a random Gaussian vector with independent random entries φiµ ∼ N (0, 1/dAdB). isapproximation will not aect the nal conclusions. en, in the notation of eorem 10.2,dAωA = dAtrB ψψ

† is the Information-Plus-Noise matrix ε2HdA , with

HdA =1

dB(RdA + σXdA) (RdA + σXdA)† , (10.16)

where

a) XdA is the dA × dB random matrix√dAdBφ, whose entries Xiν =

√dAdBφiν ∼ N (0, 1);

b) RdA is the dA × dB random matrix√dAdB |φME〉, independent of XdA ;

c) We have d−1B RnR

†n = dAtrB |φME〉 〈φME| = 1A; its ESd is clearly equal to µ(t) = δ(t− 1)

for all dA;

c) e limit limdA→∞dBdA

= c > 0 exists.

d) We have denoted σ =√

1−ε2ε .

en, according to eorem 10.2, the limiting Stieltjes transform sH(z) of HdA satises

sH(z) =

ˆdµ(t)

(1 + σ2csH(z))−1t− (1 + σ2csH(z)) z + σ2 (1− c)

=1

(1 + σ2csH(z))−1 − (1 + σ2csH(z)) z + σ2 (1− c). (10.17)

Using the scaling property (10.14), and by restoring the position σ =√

1−ε2ε , we nally get the

limiting Stieltjes transform sε2H(z) = ε−2sH(zε−2):

s =1 +

(1− ε2

)cs

ε2 − (1 + (1− ε2) cs)2 z + (1− ε2) (1 + (1− ε2) cs) (1− c). (10.18)

153

Proof of eorem 10.1. From Lemma 10.3 we found that, in the large size limit dA → ∞ (withdB = dA, i.e. c = 1 in (10.13)), the limiting Stieltjes transform of dAωA is the solution of thefollowing algebraic equation:

s =1 +

(1− ε2

)s

ε2 − (1 + (1− ε2) s)2 z. (10.19)

e Stieltjes trasform s(z) of %(λ) is dened for every z /∈ supp %. If we introduce the variableb = 1 +

(1− ε2

)s we can rewrite (10.19) as

b− 1

(1− ε2)=

b

ε2 − b2x, (10.20)

for x /∈ supp %. e inverse x = x(b) of b(x) is then

x(b) =ε2

b2− (1− ε2)

b(b− 1). (10.21)

Now the qualitative analysis predicts that the stationary points of (10.21) are the edges of supp %.An elementary computation shows that these stationary points are given by (10.6).

10.2 Inject separability in the unbiased ensemble

Let us now consider the polarization of the unbiased ensemble by a random pure separable state.is polarized ensemble reads

ψ =[εUA ⊗ UB +

√1− ε2UAB

]|φSep〉 ,

= ε |ψ′1〉 ⊗ |ψ′′1〉+√

1− ε2 |ψ2〉 (10.22)

with |ψ′1〉 ∼ Unif(HA), |ψ′′1〉 ∼ Unif(HB) and |ψ2〉 ∼ Unif(HA ⊗HB) independently dis-tributed. e typical local purity πAB(ψ) of this one-parameter ensemble with a separablepolarization is given by (9.38). What about the ESd of the rescaled eigenvalues %dAωA(λ) =d−1A

∑i δ(λ− dAλi)? e two extremal situations are now:

Unbiased ensemble For ε = 0 we have the unbiased ensemble. According to eorem 8.8

%dAωA(λ) c

2πλ

√(λ− λ−)(λ+ − λ) 1 λ−<λ<λ+ , with λ± =

(1±√c)

2

c. (10.23)

Separable ensemble For ε = 1 we are considering a random separable state. In this case

%dAωA(λ) =1

dA[(dA − 1)δ(λ) + δ(λ− dA)] , for all dA, dB, c . (10.24)

In this case then %dAωA(λ) δ(λ).

154

Here we see that the situation is more delicate since, for ε = 1 the limiting spectral density doesnot describe the spectrum of a density operator (a positive normalized matrix). For generic valuesof the oset ε ∈ (0, 1) we have again an interesting numerical evidence.

Fact 10.2 (A numerical evidence). For suciently small values of ε > 0 the support of theESd of ωA = trB ψψ

† (for the ensemble (10.22)) is compact and connected. For any xed (butlarge) dA, dB there exists a critical value ε? > 0 such that for ε > ε? the support of the ESd ofωA = trB ψψ

† ceases to be connected. In particular, the maximum eigenvalue “evaporates” fromthe “sea” of the rest of the spectrum. See gure 10.2.

How to explain this numerical evidence? e analysis of this eect can be simplied if weignore the coherent term appearing in ωA and consider instead the convex combination

σA = ε2 |ψ′1〉 〈ψ′1|+ (1− ε2)trB |ψ2〉 〈ψ2|. (10.25)

is approximation is justied in the large size limit. Indeed, using the same notation as in (9.10),standard linear algebra and a simple computation (see [54]) gives an upper bound to the averagedistance of σA from ωA

E‖ωA − σA‖22 = Etr (S2) = ε2(1− ε2)2

dB. (10.26)

en using some standard properties and Lemma 2.12 we can bound the average Levy distanceL(FωA , F σA) between the ESD of ωA and σA:

E[L(FωA , F σA)] ≤ E‖ωA − σA‖ ≤√

E‖ωA − σA‖22 =

√2ε2(1− ε2)

dB. (10.27)

By following [18], one can show that for suciently small ε the limit empirical spectraldistribution of σA is supported in a single compact interval (in the limit dA, dB → +∞, withdB/dA ∼ c > 0). When ε increases, the largest eigenvalue λmax evaporates from the sea. isphase transition occurs at a threshold value ε?(dA; c). e exact statement of this picture is thecontent of eorem 10.5 below.

In the proof of eorem 10.5 we will use the following technical result.

Lemma 10.4. Let |x〉 be a n-dimensional unit vector and A = diag(a1, . . . , an). en, the eigen-values λ1, . . . , λn of the matrix

H = A+ α |x〉 〈x| (10.28)are solutions of the secular equation

1 = α∑i∈[n]

|ψi|2

λ− ai. (10.29)

Moreover, if a1 < a2 < · · · < an and α > 0, the eigenvalues satisfy the interlacing law

a1 < λ1 < a2 < λ2 < · · · < an < λn . (10.30)

155

Proof. e secular equation det(λ−H) = 0 can be expanded as

0 = det(λ−H)

= det(λ−A) det(1− α(λ−A)−1 |ψ〉 〈ψ|)= det(λ−A)

[1− αtr ((λ−A)−1 |ψ〉 〈ψ|)

]= det(λ−A)

1− α∑i∈[n]

|ψi|2

λ− ai

. (10.31)

In the third line we have took into account that (λ−A)−1 |ψ〉 〈ψ| has rank 1. en, the secularequation for H is satised when the term in square brakets vanishes. It is easy to see that thezeroes of det(λ−A) do not contribute thanks to the poles ai’s in the second factor. e interlacingproperty is a standard result in linear algebra.

eorem 10.5. Let us consider the ensemble of density matrices (10.25).For large dA and dB ∼ cdA,there exists a value ε?(dA; c) that separates two regimes of the EDS of dAσA. For ε < ε?, the spectralmeasure of dAσA in (10.25) is entirely supported in the interval[

(1−√c)2(1− ε2)

c,(1 +

√c)2(1− ε2)

c

]. (10.32)

When ε > ε?, on average the largest eigenvalue of dAσA detaches from the interval (10.32). ecritical value ε? = ε?(dA; c) is

ε2? =1 +√c

1 +√c+ cdA

. (10.33)

e evaporated eigenvalue of σA occurs at the position

λmax(dAσA) =ε4(cdA − c+ 1) + ε2(c− 1)

ε2(1 + cdA)− 1, for ε > ε? . (10.34)

Proof. It is convenient to work in the eigenbasis of ω2 = trB |ψ2〉 〈ψ2|. In this basis, dAω2 =dAdiag(a1, . . . , αdA). We write dAσA = (1− ε2)H and we are dealing with the following rank-1perturbation problem

H = dA

[ω2 +

ε2

1− ε2|φ〉 〈φ|

], (10.35)

where we have renamed |ψ′1〉 ≡ |φ〉 for saving notation. Using Lemma 10.4 the eigenvalues dAλi’sof H are solution of the following equation

1 =ε2

1− ε2E∑i∈[dA]

dA|φi|2

dAλ− dAai, (10.36)

and satisfy the interlacing property a1 < λ1 < a2 < λ2 < · · · < adA < λdA . We are interestedin the position of the largest eigenvalue λmax ≡ λdA of H (which is not trapped by the ai’s) as a

156

function of ε and dA. In the large dA limit, the continuous version of (10.36) is easily obtained byaveraging over |φ〉 and using the limit density %(x) (see eorem 8.8) for dAai.

1 =ε2

1− ε2E[dA|ψi|2

]dA

ˆdy

E[%dAω2(y)]

dAλ− y, (10.37)

where we used the independence of |φ〉 and ω2. Here we have E[|φi|2

]= 1

dAfor all i ∈ [dA] and

the limiting ESd of dAω2 is

%(y) =c

2πy

√(y − y−)(y+ − y) , with y± =

(1±√c)2

c. (10.38)

by eorem 8.8. en we seek for a solution dAλ ≥ (1 +√c)2/c of

1 =ε2

1− ε2dA

ˆdx

E[%(y)]

dAλ− x. (10.39)

e transformation yc = x makes the problem easier:

1 =ε2

1− ε2dA

ˆ y+

y−

dy

dAλ− yc

2πy

√(y − y−)(y+ − y)

=ε2

1− ε2dA

ˆ x+

x−

dx

dAλ− xc

1

2πx

√(x− x−)(x+ − x)

=ε2

1− ε2cdA

ˆ x+

x−

dx

cdAλ− x%c(x)

= − ε2

1− ε2cdAs%c(cdAλ) (10.40)

Here we have denoted by %c(x) = (2πx)−1√

(x− x−)(x+ − x) with x± = (1 ±√c)2 the

standard Marcenko-Pastur law with parameter c ≥ 1 and by s%c(z) its Stieltjes transform. eStieltjes transform of the Marcenko-Pastur law is given by (2.133):

s%c(z) =(c− 1)− z +

√(x− − z)(x+ − z)2z

. (10.41)

Remind that we are always supposing dAλ > y+, i.e. outside the support of the “sea” of eigenvalues(this means that we are considering the largest eigenvalue of dAσA). en, we end up with

1 =ε2cdA1− ε2

cdAλ− (c− 1)−√

(x− − cdAλ)(x+ − cdAλ)

2cdAλ. (10.42)

e right-hand side of (10.42) is monotonically decreasing for cdAλ ≥ x+. At cdAλ = x+ thesquare root vanishes and this maximum value is ε2cdA

1−ε21

1+√c. en, in order for (10.42) to have a

solution cdAλ > c+ we must have

ε2cdA1− ε2

1

1 +√c> 1 . (10.43)

157

is condition translates into ε > ε?, with the critical value ε? as in (10.33). Solving this equationfor ε > ε?, and taking into account the multiplicative factor (1− ε2) we nally get the location ofthe detached eigenvalue (10.34).

See Figure 10.2 for a numerical check of these results.

1

N

D = 0

D = 0.01 D = 0.05

Λ

ΡHΛ

L

Figure 10.2: Comparison between the numerical simulation and the analytic results for the evaporation phenomenonof the ensemble (10.25). We have generatedN = 105 random states σA of the ensemble (10.25) and we have numericallydiagonalized them. We have denoted ∆ = ε2 − ε2?, where ε? is the critical value of the tunable parameter given byEq. (10.33). e theory predicts that for positive values of ∆, the largest eigenvalue of σA becomes on outlier andevaporates form the bulk of %ωA . In the regime ∆ > 0 the largest eigenvalue is located at the position given by (10.34).e solid lines in gure are the smooth kernel estimators of the sampled eigenvalues (in Mathematica denoted asSmoothHistogram[ ]), for three dierent values of ∆ ≥ 0. e points on the axes indicates the predicted position ofthe largest eigenvalue, Eq. (10.34). e agreement with the numerical estimators is excellent. In the gure dA = 100,and c = 3.

We conclude by mentioning that the problem of the spectral measure of the sum of randommatrices has been intensively investigated in the framework of free probability. It is well knownthat the eigenvalues of the sum of two n × n self-adjoint matrices An and Bn depends in acomplicated way not only on the spectra of An and Bn but also on the relative orientations oftheir eigenvectors. Nevertheless, if one add some randomness to the eigenspaces so that theyare in generic position with respect to each other, in the large n limit, free probability (see Section2.4) provides a good understanding of the spectral measure of their sum. Roughly speaking, forlarge dimensional (independent) random matrices An and Bn, the spectral measure of the sumAn + Bn can be determined solely from the eigenvalue densities of the individual matrices. IfAn → A and Bn → B in the sense of moments, for n → ∞, then the spectrum of the sumAn +Bn converges to the free additive convolution AB of the individual limits [204].

e study of the inuence of nite rank additive perturbations on the asymptotic spectrum ofclassical random matrix models has aracted a lot of aention in the last years. In the literature theproblem of the evaporation of the extremal eigenvalues from the sea that accomodates the majority

158

of the spectrum is known as problem of the outliers for spiked additive deformations of classicalrandom matrices. e outliers, i.e. the eigenvalues that deviate from the support of the bulk, are“generated” by the “spikes”, i.e. the eigenvalues of the perturbation. A new understanding of theproblem has been achieved when some authors realized that the phenomenon can be consistentlydescribed in free probabilistic terms through the so-called free subordination property. Solvingthe problem of the outliers consists in solving an equation involving the free subordination mapand the spikes of the perturbation. is method provides a unied approach for the additivepertubation of classical random matrix models as well as the Information-Plus-Noise models. isvery interesting approach and the consequent ramications goes beyond the present goal and werefer to the works [42, 43] and references therein for a good survey.

10.3 Superposition of maximally entangled states

As a last example of polarized ensemble, let us consider the coherent superposition of twoindependent random maximally entangled state. Without loss of generality, we can considerthe superposition of a xed maximally entangled state |φME〉 with another random maximallyentangled state:

|ψ〉 =1√C

[ε1AB +

√1− ε2(UA ⊗ 1B)

]|φME〉 . (10.44)

Here 1AB and 1B denote the identity operator on HA ⊗HB and HA respectively. e localunitary operator UA ∈ U(HB) is distributed according to Haar measure. e parameter ε variesbetween 0 and 1 and C is a normalization constant. For this specic ensembles we are able towrite explicitly the spectral density of the reduced density matrix.

eorem 10.6. Let us consider the random state (10.44) as above with dA ≤ dB . e ESd of therandom matrix dAωA

%dA(λ) =1

dA

∑i∈[dA]

δ(λ− dAλi) (10.45)

weakly converges as dA →∞ to the the arcsine law supported in [λ−, λ+]:

%ε(λ) =1

π

1√(λ− λ−)(λ+ − λ)

, with λ± = 1± 2ε√

1− ε2 . (10.46)

e average purity πAB(ψ) of the ensemble is

E[πAB(ψ)] =1 + 2ε2 − ε4

dA. (10.47)

Proof. By tracing over HB , the reduced state on HA is

ωA =1A + ε

√1− ε2(UA + U †A)

dA

[1 + tr (UA + U †A)

] , (10.48)

159

where we have used trB |φME〉 〈φME| = 1A/dA Since the unitary operatorUA is Haar distributed,the phases αi of the eigenvalues eiαi of UA are uniformly distributed on the unit circle

αi ∼ Unif(0, 2π) , for all i ∈ [dA] . (10.49)

erefore, in the large dA limit the term tr (UA + U †A) in the normalization can be neglected. Itreadily follows that the eigenvalues of the reduced density matrix ωA have the form

λi =1 + 2ε

√1− ε2 cosαidA

, i ∈ [dA] . (10.50)

en, (10.46) follows from the elementary change of variables (10.50) from αi to λi. Knowing thelimiting spectral density, a standard computation gives the average (10.47) at leading order.

Remark 10.3. e alert reader will notice that we have presented a “proof” and not a “strict proof”.“Proofs” are enough at a phenomenological level and we feel that the eort needed to presenta strict proof is not worth suciently for the purposes of this Chapter . A strict proof of theconvergence of the ESd (10.45) is possible using the devices of Free probability theory, and inparticular theR-transform for the additive free convolution (see Denition 2.5). We le the detailsfor the reader.

Our polarized superposition of maximally entangled states is a generalization of the “arcsineensemble” studied by Zyczkowski et al. [241]. In [241], the authors considered the symmetricsuperposition (ε = 1/

√2) of two maximally entangled states in the balanced bipartite Hilbert

space (dA = dB). We stress the fact that the limiting spectral density is independent on theasymmetry of the bipartition as long as dA ≤ dB .

For a numerical verication of the convergence of the EDS (10.48) and the average purity(10.47) of the ensemble see gure 10.3.

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ççç

Ε = 0.5

Ε' = 0.25

0 Λ'- Λ'

-Λ- Λ+1 2

Λ

ΡHΛL

0.0 0.2 0.4 0.6 0.8 1.01.0

1.1

1.2

1.3

1.4

1.5

Ε2

d AE

@Π AB

D

Figure 10.3: Comparison of numerical simulations and analytic expression of the rescaled ESd and the typical purityof the generalized arcsine ensemble (10.44). Top. e analytic curves are the limiting spectral densities (10.46) forε = 1/4 (magenta) and ε = 1/2 (blue). e points come from numerical simulations with dA = 50 and dB = 70. esample size is N = 103. Bottom: e analytic curve is given by (10.47). Notice that the curve is symmetric withrespect to the balanced superposition ε = 1/

√2. e points are numerical simulation of the polarized ensemble with

dA = 50, dB = 70 and several values of ε. e sample size for each point is N = 103.

Chapter 11

Random Tensor eory

We consider in this last Chapter another problem in RMT inspired by quantum informationtheory. In a recent work, Ambainis, Harrow and Hastings [6], inspired by questions formulatedby Leung and Winter [141], considered an interesting extension of RMT to Random Tensor eory,namely from random matrices to random mixture of tensor product states. In this Chapter, wewill review old (Marcenko-Pastur) and new (Ambainis-Harrow-Hastings) scenarios in a uniedframework and we will prove a “macroscopic universality“ of the Marcenko-Pastur law. Furtherinvestigations are needed and we hope that this contribution could trigger the interest of themathphysical community.

11.1 Setting

We shall consider as acting in a n-dimensional Hilbert space H a self-adjoint operatorW of theform

W =∑µ∈[m]

ψµ ∈ B(H ) . (11.1)

e operatorW is a sum of m random rank-1 projections ψµ ≡ |ψµ〉 〈ψµ| (µ ∈ [m]), where the|ψµ〉 are i.i.d. unit vectors whose law will be specied later. It is immediate to verify thatW ≥ 0and trW = m. e “natural” domain H ofW (that is to say the “natural” choice of n) will beclear below. Hereaer H will be a d-dimensional Hilbert space. We will endow the unit sphereS(H) with the (unique) unitarily invariant measure. A random unit vector distributed accordingto this probability law will be denoted as ψ ∼ Unif(H).

We will analyse dierent laws for the unit vectors |ψµ〉. Let us consider the following scenarios:

(MP) In the original seing [154], Marcenko and Pastur considered

|ψµ〉 ∼ Unif(H) . (11.2)

In this case it is natural to take H ≡ H as natural domain of W and n ≡ d. We will call itthe “Marcenko-Pastur ensemble” (or (MP) for short).

161

162

(AHH) Ambainis, Harrow and Hastings [6] considered a tensor product structure

|ψµ〉 = |ψ1µ〉 ⊗ |ψ2

µ〉 ⊗ · · · ⊗ |ψkµ〉 , with |ψξµ〉 ∼ Unif(H) i.i.d. . (11.3)

In order to identify the natural domain of W , we observe that

span|ψ1µ〉 ⊗ |ψ2

µ〉 ⊗ · · · ⊗ |ψkµ〉 : |ψξµ〉 ∈ H , ∀ξ ∈ [k]

= H⊗k . (11.4)

is observation immediately suggests to take H ≡ H⊗k and n ≡ dk. We will call thisrandom tensor ensemble the “Ambainis- Harrow-Hastings ensemble” (or (AHH) for brevity);

(gAHH) A natural generalization of the (AHH)-ensemble is

|ψµ〉 = |ψ1µ〉 ⊗ |ψ2

µ〉 ⊗ · · · ⊗ |ψkµ〉 , with |ψjµ〉 ∼ Unif(H(j)) i.i.d. (11.5)

Here dimH(j) = dj , for j ∈ [k] and then H =⊗

j∈[k] H(j), n ≡

∏j∈[k] dj . is ensemble

will be denoted as (gAHH) (g standing for “generalized”);

(F) An interesting ensemble for applications in quantum information arises by taking

|ψµ〉 = |ϕµ〉⊗k , |ϕµ〉 ∼ Unif(H) . (11.6)

Each random vector in |ψµ〉 ∈ H⊗kis the k-fold tensor product of an elementary randomvector in H. We observe that

span|ϕµ〉⊗k : |ψjµ〉 ∈ H , ∀j ∈ [k]

= Hk , (11.7)

where Hk is the symmetric subspace of H⊗k , k > 1. en W naturally acts on H ≡ Hk

and then n ≡(d+k−1k

), the dimension of Hk. is ensemble will be called the “Fold”

ensemble (abbreviated as (F));

(Sym) e above ensemble suggests to consider also the case of unit vectors uniformly distributedon the k-symmetric subspce

|ψµ〉 ∼ Unif(Hk) , (11.8)

with again H ≡ Hk and n ≡(d+k−1k

);

(Anti) If we denote by H∧k the antisymmetric subspace of H⊗k , k > 1, a counterpart of the previosensemble is

|ψµ〉 ∼ Unif(H∧k) . (11.9)

In this case, since the random vectors of the ensemble span the antisymmetric space whosedimension is

(dk

)we have H ≡ H∧k and n ≡

(dk

);

163

Wn,c =∑

µ∈[m] ψµ |ψµ〉 n

(MP) |ψµ〉 ∼ Unif(H) d

(AHH) |ψµ〉 =⊗

ξ∈[k] |ψξµ〉 , |ψξµ〉 ∼ Unif(H) dk

(gAHH) |ψµ〉 =⊗

ξ∈[k] |ψξµ〉 , |ψξµ〉 ∼ Unif(Hξ)

∏ξ∈[k] d

kξ

(F) |ψµ〉 = |ϕµ〉⊗k , |ϕµ〉 ∼ Unif(H)(d+k−1k

)(Sym) |ψµ〉 ∼ Unif(Hk)

(d+k−1k

)(Anti) |ψµ〉 ∼ Unif(H∧k)

(dk

)(Perm) |ψµ〉 ∼ Unif(Sd(|φ0〉)) , |φ0〉 ∈ H depends on |φ0〉

Table 11.1: Summary of the notation used in the Chapter.

(Perm) It would be interesting to consider also the case where |ψµ〉 comes from a nitely supportedprobability law. Here we will investigate

|ψµ〉 ∼ Unif(Sd(|φ0〉)) , (11.10)

i.e. uniformly over the orbit of the permutation group Sd through a ducial vector |φ0〉 ∈ H.We shall see that it is pertinent to take H ≡ H and n ≡ d.

In (11.1) the number of summands (the number of |ψµ〉’s) is m = m(n). We will always assumethat the following limit exists

c = limn→∞

m

n∈ ]0,+∞[ . (11.11)

Marcenko and Pastur called this limit the “concentration” of the ensemble [154]. In order to keeptrack of this parameter we will denote the operator (11.1) asWn,c.

During the rest of the Chapter we will use this notation. Before starting we recall a well-knownresult.

eorem 11.1 (Marcenko-Pastur law). Let c ∈ ]0,+∞[. Let Xc be a real random variable withmoments

E[X`c

]=∑s=1

cs Nar(`, s) , ` ≥ 1 , (11.12)

where the Narayana numbers Nar(`, s) are

Nar(`, s) =1

s

(`

s− 1

)(`− 1

s− 1

), 1 ≤ s ≤ ` . (11.13)

en law(Xc) is given by

%c(λ) = max 0, (1− c) δ(λ) +

√(λ− λ−) (λ+ − λ)

2πλ1λ∈(λ−,λ+) , (11.14)

164

with λ± = (1±√c)

2. We will say that law(Xc) is the Marcenko-Pastur law of parameter c:law(Xc) ∼MP(c).

Proof. e proof of the theorem proceeds in two steps. Firstly, by direct computation one showsthat the moments of (11.14) are given by (11.12). en, one shows that the sequence of moments(11.12) satises the Carleman condition∑

`≥1

E[X2`c ]−1/2` = +∞ , (11.15)

and then determines the law (11.14) uniquely. Since E[X2`c

]−1/2`> 1/λ+ > 1 for all c > 0, the

condition (11.15) is automatically satised.

In the rest of the Chapter we will always use the notation λ± = (1±√c)

2.Let us come back to our self-adjoint operatorWn,c. In a xed basis |I〉I∈[n] of the Hilbert

space H we can write the unit vectors ψµ as

|ψµ〉 =∑I∈[n]

ψµI |I〉 , with∑I∈[n]

|ψµI |2 = 1 , for allµ ∈ [m] . (11.16)

Let us introduce the matrix Ψ with matrix elements given by

Ψ = (ψIµ)I∈[n],µ∈[m] with ψIµ ≡ ψµI . (11.17)

In other words, the m columns of Ψ are the m vectors |ψµ〉. In this notation, the operatorWn,c isnothing but

Wn,c = ΨΨ† . (11.18)

Random matrices of this kind were rst considered by Wishart [237] in connection to someproblems in multivariate statistics (see Sections 1.3.3-5.1 and Example 2.3).

We shall be interested in the empirical spectral density (ESd) and the empirical spectraldistribution (ESD) ofWn,c:

%n,c(λ) =1

n

n∑i=1

δ (λ− λi(Wn,c)) , (11.19)

Fn,c(t) =# λi(Wn,c) ≤ t

n=

ˆ t

−∞dλ %n,c(λ) . (11.20)

Of particular interest is the case of very large n and m, since it oen happens that for n→∞the random measure (11.19) converges in probability to a nonrandom probability measure (or,equivalently, Fn,c convergences to a nonrandom function on its continuity points). In the (MP)scenario, Marcenko and Pastur proved the convergence of the ESD ofWn,c for large n. Later,this result has been proved again several times using almost all the techniques of RMT (see thediscussion on the Wishart-Laguerre ensemble in Section 2.6). Here we stress the fact that for

165

all the other ensembles listed in Table 11.1, the random matrixWn,c has neither independententries nor is unitarily invariant. is type of self-adjoint ensemble escape from the fundamentalclassication of the classical RMT ensemble detailed in Section 1.2. en, as already argued in [6]the standard tools of RMT cannot be used and one needs to develop new (or modied) methodsin order to analyse these mixture of random tensors.

11.2 Summary of techniques and results

is is the longest Chapter of the thesis. erefore a short paragraph with a recap of the newresults, how they were obtained and the limitations of each technique should help the readerform a clearer mental picture of what to expect.

In Section 11.3 we will review the diagrammatic proof of the Marcenko-Pastur law for the(MP)-ensemble. e diagrammatic approach is perhaps the simplest way to get the strongestresult, i.e. the convergence with probability 1. Aer a preliminary Gaussianization and truncationtricks 11.3.1, one applies the moment method introduced in 2.1. e method consists essentiallyin the proof of Lemmas 11.5 and 11.6 within a diagrammatic approach based on bipartite graphs.e proof of some of the technical combinatorial Lemmas in 11.3.3 can be found in [12].

e new result in Section 11.3 is the almost sure convergence to the Marcenko-Pastur law ofthe (AHH)-ensemble and its generalization (gAHH) stated in eorem 11.11. is is a strengtheningof eorem 11.10 of Ambainis, Harrow and Hastings [6] and proves their conjecture 11.1. eproof is based on an extension of the diagrammatic method from bipartite to multipartite graphs.

e moment method is quite powerful but requires some smart way to identify the “adequate”combinatorial objects to deal with. en, one should manage to arrange and classify diagramsaccording to their weight and cardinality in the asymptotic limit. In order to show the macroscopicuniversality of the Marcenko-Pastur law for more general random tensors ensembles we havedeveloped in Section 11.5 a dierent method based on Lie groups actions and free probabilityarguments. In this language, Table 11.1 is conveniently translated in Table 11.2.

Section 11.7 introduces the problem of the universality of the Marcenko-Pastur law for randomtensors in terms of the Free Poisson law for the sum of free independent rank one projectors.e main result of this Section are eorems 11.15-11.16-11.17 on the asymptotic freeness orrandom local unitary operators (random local unitaries on multi-partite systems). e generalproof of the Free Poisson law 11.18 provides our main results (eorems 11.19-11.20 and 11.19)thus proving the Marcenko-Pastur law as macroscopic limit for a large class of random tensors(all the ensembles in Table 11.1).

It is worth mentioning another generalization of the classical (MP)-ensemble. Pajor andPastur [181] studied the macroscopic limit of the sum of random rank one projectors determinedby i.i.d. isotropic vectors with log-concave probability law. Using some fundamental results aboutlog-concave measures they proved the convergence in probability to the Marcenko-Pastur law (wedo not know if it is possible to get the almost sure convergence within this approach). Howeverthis technique is not suitable for our purposes. e random vectors in Table 11.1 have probabilitylaw supported on the extremal points of a convex set (points on the sphere) and therefore the

166

log-concave condition is not valid.At the end of the Chapter we oer two remarks (Section 11.9) on the implications of our

results in Random Matrix eory. First, we give the matrix representations of the various randomtensor scenarios. In this perspective, mixtures of random tensors are covariance matrices of datasets with an underlying paern in the entries. is is a departure from the classical paradigm instatistics. Second, we elaborate more on the concept of “generic position” of random matrices andthe law of addition for random locally rotated operators.

11.3 e Marcenko-Pastur law

In the scenario (MP) it is natural to put n ≡ d and considerWn,c as acting on the d-dimensionalspace H. We will call such aWn,c a (MP)-matrix. Marcenko and Pastur proved the followingtheorem

eorem 11.2 (Marcenko and Pastur [154]). LetWn,c be a (MP)-matrix. With the identicationn ≡ d, for n→∞, the ESd ofWn,c converges weakly in probability to theMP(c) law

%n,c(λ) %c(λ) , as n→∞ . (11.21)

Later a stronger convergence has been proved [238].

eorem 11.3. e conclusion of the previous theorem remains true when convergence in proba-bility is replaced by almost sure convergence.

Before turning to the actual proof, it is useful to apply a preliminary operation, namely theGaussianization and the truncation tricks.

11.3.1 Gaussianization and Truncation tricks

For the Gaussianization trick the following lemma will be precious.

Lemma 11.4 (Property of the chi-square variables). Let r ∼ χ2k a standard chi-square variable

with k degrees of freedom. en:

(i) e moments of r are

E[r`] = 2`Γ(`+ k

2 )

Γ(k2 ). (11.22)

(ii) For the square root of a standard chi-square variable the following asymptotic result holds√

2r ∼ AN (√

2k − 1, 1) . (11.23)

167

Proof. e law of r is

fr(x) =xk−1e−x

Γ(k). (11.24)

From a classical computation (see for instance Section 7.5 o [231]) one knows explicitly thecharacteristic function of r and therefore its moments (11.22). For the second part of the Lemma,we observe that the law of

√2r is

f√2r(y) =y2k−1e−y

2/2

2k−1Γ(k). (11.25)

A saddle-point computation shows that the maximizer of (11.25) is y? =√

2k − 1 and the secondderivative df√2r(y

?)

dy = −2. is shows the asymptotic normality (11.23).

With this concepts in mind we can now apply the Gaussianization trick and the truncationtrick to the (MP)-matrixWn,c.

Gaussianization We have already observed in Section 8.2 that a d-dimensional random vectorψµ =

(ψ1µ, ψ2µ, . . . , ψdµ

)in H with i.i.d. Gaussian components ψiµ ∼ NC(0, 1/d) is exponen-

tially close to the unit sphere S(H). Indeed, for all µ ∈ [m] let us write ψµ =√rµ |ψµ〉 where

the rµ’s are indpendent chi-square random variables with 2d degrees of freedom rµ ∼ χ22d/2d.

Let us dene

Ψ =(ψiµ

)i∈[d],µ∈[m]

(11.26)

Wn,c = ΨΨ† . (11.27)

Let us denote the ESD’s of Wn,c and Wn,c as FWn,c(λ) =´ λ−∞ dx %Wn,c(x) and FWn,c

(λ) =´ λ−∞ dx %Wn,c

(x) respectively. By Lemma 2.13 we are able to bound the Levy’s distance( see

168

Section 2.5)

L4(FWn,c , FWn,c

)≤ 2

n2

[tr (ΨΨ† + ΨΨ†)

] [tr (Ψ− Ψ)(Ψ− Ψ)†

]≤ 2

n2

∑i∈[d],µ∈[m]

(|ψiµ|2 + |ψiµ|2)

∑i∈[d],µ∈[m]

(|ψiµ − ψiµ|2)

≤ 2

n2

∑i∈[d],µ∈[m]

(1 + rµ)|ψiµ|2 ∑

i∈[d],µ∈[m]

(1−√rµ)2|ψiµ|2

≤ 2

n2

∑i∈[d],µ∈[m]

(1 + rµ)

∑i∈[d],µ∈[m]

(1−√rµ)2

≤ 2

∑µ∈[m]

(1 + rµ)

∑µ∈[m]

(1−√rµ)2

−→ 0 , (11.28)

where, using Lemma 11.4, the last line follows from the law of large numbers.

Truncation Let C be a positive number and dene

ψiµ = ψiµ 1|ψiµ|<C (11.29)

Ψ = (ψiµ)i∈[d],µ∈[m] (11.30)

Wn,c = ΨΨ† . (11.31)

By virtue of Lemma 2.13 we can control the Levy’s distance between FWn,cand FWn,c

L4(FWn,c

, FWn,c

)≤ 2

n2

[tr (ΨΨ† + ΨΨ†)

] [tr (Ψ− Ψ)(Ψ− Ψ)†

]≤ 2

n2

∑i∈[d],µ∈[m]

(|ψiµ|2 + |ψiµ|2)

∑i∈[d],µ∈[m]

(|ψiµ − ψiµ|2)

≤ 2

n2

2∑

i∈[d],µ∈[m]

(|ψiµ|2)

∑i∈[d],µ∈[m]

(|ψiµ|2 1|ψiµ|>C

−→ 4cE

(|ψiµ| > C

), (11.32)

where the last line follows again from the law of large numbers. We note that the right hand sidecan be made arbitrarily small by choosing C large enough.

169

Combining together (11.28) and (11.32) we may assume that the variables ψiµ are i.i.d. withtruncated Gaussian law. For abbreviation, in proofs given in the next steps, we still useW , Ψ andψiµ for the quantities associated to the truncated variables.

For later convenience it will be also useful to work with standardized variables. In order todo that, from now on we assume that ψiµ are i.i.d. variables uniformly bounded with mean zeroE[ψiµ] = 0 and unit variance E

[|ψiµ|2

]= 1, and we shall consider the following operator

Wn,c =1

nΨΨ† . (11.33)

11.3.2 e moment method for Wishart matrices

One of the most popular technique in RMT is the moment method, which uses the momentconvergence theorem (see Section 2.1). As noticed before, the `-th moment of the ESd ofWn,c canbe wrien as ˆ

dλλ`%n,c(λ) =1

ntrW`

n,c . (11.34)

is expression plays a fundamental role in RMT. We will show that the sequence n−1trW`n,c

converges to the sequence of moments ofMP(c). By applying the moment convergence theoremwe complete the proof of the Marcenko-Pastur law by showing the following lemmas:

Lemma 11.5. For all ` ≥ 1:

limd→∞

E

[1

ntrW`

n,c

]=∑r=1

cr Nar(`, r) . (11.35)

Lemma 11.6. For all ` ≥ 1:

Var

(1

ntrW`

n,c

)= O(n−2) . (11.36)

Proof of eorem 11.3. Assume that Lemmas 11.5 and 11.6 have been proved. To conclude theproof of eorem 11.3, one needs to check that for any bounded continuous function ϕ

limn→∞

〈%n,c , ϕ〉 = 〈%c , ϕ〉 , a.s. , (11.37)

with %c(λ) theMP(c) law (11.14). Using (11.35), the moment convergence theorem [12] and theprevious eorem 11.1, one proves the convergence in expectation

limn→∞

E 〈%n,c , ϕ〉 = 〈%c , ϕ〉 . (11.38)

To prove the almost sure convergence (11.37), we use the estimate on the variance (11.36). Indeed,by Chebychev’s inequality, for any δ > 0:

Pr

(∣∣∣∣ 1n [trW`n,c

]− 1

nE[trW`

n,c

]∣∣∣∣ > δ

)≤ O

(1

n2

), ∀` , (11.39)

170

and the Borel-Cantelli lemma implies

limn→∞

∣∣∣∣ 1n [trW`n,c

]− 1

nE[trW`

n,c

]∣∣∣∣ = 0 a.s. . (11.40)

e starting point of the proofs of Lemmas 11.5 and 11.6 relies on (11.34):ˆdλλ`E[%n,c(λ)] =

1

nE[trW`

n,c]

= d−`−1∑

µ1,...,µ`∈[m]

∑i1,...,i`∈[d]

E[ψi1µ1ψi2µ1ψi2µ2ψi3µ2 · · ·ψi`µ`ψi1µ`

](11.41)

To use the moment method we need to recall some lemmas on graph theory and combinatorics.

11.3.3 Some lemmas on graph theory and combinatorics

Usually, the moment method is implemented by establishing a bijection between “words” whoseleers are the random entries of a random matrix ensemble and combinatorial objects. ediagrammatic that we are going to use is inspired by Bai and Silverstein [12]. Although the resultswe are going to present are classical, we believe that a quick review will serve as a training to dealwith the random tensor mixture (see later). Indeed, we will show in the next sections that thesame diagrammatic developed for classical Wishart matrices is well-tailored for random tensorsensembles.

Let use introduce the bipartite graphs G(i,µ), where i = (i1, . . . , i`) ∈ [d]` and µ =(µ1, . . . , µ`) ∈ [m]` are the (not necessarly distinct) vertices of the graph. e “down” edges ofthe graph are (ia, µa), a ∈ [m] and the “up” edges are (µa, ia+1), a ∈ [`], with the conventionthat i`+1 = i1 (see Fig. 11.1). In the following we will denote

ψG(i,µ) ≡ ψi1µ1ψi2µ1ψi2µ2ψi3µ2 · · ·ψi`µ`ψi1µ` , (11.42)

where i = (i1, . . . , i`) and µ = (µ1, . . . , µ`). Two graphs G1(i1,µ1), G2(i2,µ2) are said to beisomorphic if there exist two permutations σ ∈ Sd, π ∈ Sm such thatG2(i2,µ2) = G1(σi1, πµ1),i.e. if G2 can be obtained form G1 by a relabelling of the vertices. For each isomorphism class,there is only one graph, called canonical, satisfying i1 = µ1 = 1, ia ≤ maxi1, . . . , ia−1 + 1and µa ≤ maxµ1, . . . , µa−1+ 1. e canonical graph of the class [G(i,µ)] will be denoted by∆(`, r, s) ifG(i,µ) has r+1 noncoincident i-vertices and s noncoincidentµ-vertices (see Fig.11.1).Obviously, for each m, r and s, the number of graphs in the isomorphism class represented bythe canonical ∆(`, r, s)-graph is

#[∆(`, r, s)] = m(m−1) . . . (m−(s−1))d(d−1) . . . (d−r) = msdr+1(1 +O(d−1)

). (11.43)

We classify ∆(`, r, s)-graphs into three main classes (see Figure 11.2):

171

i1 = i4

µ1 µ2

i2 = i3

µ3

G(i,µ)

i

µ

ψG(i,µ) = ψi1µ1ψi2µ1ψi2µ2ψi2µ2ψi2µ3ψi1µ3

1

1 2

2

3

∆(`, r, s)

i

µ

ψ∆(`,r,s) = ψ11ψ21ψ22ψ22ψ23ψ13

Figure 11.1: Example of bipartite graph G(i,µ) and its canonical representant ∆(`, r, s). e word ψG(i,µ) is on thele of the graph. Here i = (i1, i2, i3, i4 = i1) and µ = (µ1, µ2, µ3). In the example, ` = 3, the number of distinct i(up) vertices is r + 1 = 2 and the number of distinct µ (down) vertices is s = 3. For pictorial convenience, we usebend arrows for the “down edges”.

• ∆1(`, s): the ∆(`, r, s)-graphs in which each down edge concides with one and only oneup edge. In this category r + s = ` and thus we suppress the r dependence;

• ∆2(`, r, s): the ∆(`, r, s)-graphs that contain at least one single edge;

• ∆3(`, r, s): ∆(`, r, s)-graphs that do not belong to ∆1(`, r) or ∆2(`, r, s).

We have the following lemmas:

Lemma 11.7. For m and s xed the number of ∆1(`, s)-graphs is

|∆1(`, s)| = Nar(`, s)

=1

s

(`

s− 1

)(`− 1

s− 1

). (11.44)

Moreover Eψ∆1(`,s) = 1.

Proof. For a proof of (11.44) see Lemma 3.4 of [12]. Since the ψiµ’s are i.i.d. with E|ψiµ|2 = 1,we have:

Eψ∆1(`,s) = E∏t∈[s]

|ψtt|2 =∏t∈[s]

E|ψtt|2 = 1. (11.45)

Lemma 11.8. For every ∆2(`, r, s): Eψ∆2(`,r,s) = 0.

172

1

1

2

2

i

µ

∆1(3, 2) = |ψ11|2|ψ21|2|ψ22|2

1

1 2

2

3

i

µ

∆2(3, 1, 3) = ψ11ψ21|ψ22|2ψ23ψ13

1

1

2

2

i

µ

∆3(4, 1, 2) = |ψ11|2|ψ21|2|ψ22|4

Figure 11.2: Examples of canonical graphs ∆(`, r, s).

Proof. e proof is trivial, since the word ψ∆2(`,r,s) contains at least one nonpaired centeredrandom variable.

Lemma 11.9. e total number r + 1 + s of noncoincident vertices of a ∆3(`, r, s)-graph is lessthan or equal to `:

r + s < ` , for every ∆3(`, r, s)-graph . (11.46)

Proof. For a proof see Lemma 3.3 of [12].

We can now turn to the proof of Lemmas 11.5 and 11.6.

Proof of Lemma 11.5. We wish to compute the average moments of

Wn,c =1

nΨΨ† , Ψ = (ψiµ)i∈[d], µ∈[m] , (11.47)

where

ψiµ =ψiµ√

E[|ψiµ|2 1|ψiµ|<C ]1ψiµ

(11.48)

withψiµ

i.i.d.∼ NC(0, 1) . (11.49)

173

In particular we have that the ψiµ’s are i.i.d. with

|ψiµ| < C , Eψiµ = 0 , E|ψiµ|2 = 1 . (11.50)

We recall once again that for the (MP)-ensemble we identify n ≡ d. e `-th moment (here ` isxed) is given by

1

nE[trW`

n,c

]=

1

dE[trW`

n,c

]= d−`−1

∑µ1,...,µ`∈[m]

∑i1,...,i`∈[d]

E[ψi1µ1ψi2µ1ψi2µ2ψi3µ2 · · ·ψi`µ`ψi1µ`

]= d−(`+1)

∑i,µ

EψG(i,µ)

(by summing over the isomorphism classes and using (11.43))

= d−(`+1)∑

∆(`,r,s)

msdr+1[1 +O(d−1)]Eψ∆(`,r,s)

(split the sum according to ∆1(`, s), ∆2(`, r, s) and ∆3(`, r, s))

= d−`

∑∆1(`,s)

msdrEψ∆1(`,r) +∑

∆2(`,r,s)

msdrEψ∆2(`,r,s) +∑

∆3(`,r,s)

msdrEψ∆3(`,r,s)

+O(d−1)

(by using limm

d→ c)

= d−`

∑∆1(`,s)

csd`Eψ∆1(`,r) +∑

∆2(`,r,s)

csdr+sEψ∆2(`,r,s) +∑

∆3(`,r,s)

csdr+sEψ∆3(`,r,s)

+O(d−1)

(using Lemmas 11.7-11.8-11.9)

=∑

∆1(`,s)

cs +∑

∆2(`,r,s)

0 +∑

∆3(`,r,s)

csO(d−1)Eψ∆3(`,r,s)

(summing on all distinct isomorphism classes of type ∆1 and using 11.7)

=∑s=1

cs|∆1(`, s)|+O(d−1) −→∑s=1

cs Nar(`, s) .

174

Proof of Lemma 11.6. Since the entries ψiµ of Ψ are uniformly bounded by a constant C , one canshow that

Var

[1

ntrW`

n,c

]=

1

d2

E[(trW`

n,c

)2]−E

[trW`

n,c

]2

= d−2d−2`∑i,j,µ,ν

E[ψG1(i,µ)ψG2(j,ν)]−EψG1(i,µ)EψG2(j,ν)

≤ K`C

2`d−2 ,

where K` depends only on `. Indeed, if G1 has no edges coincident with edges of G2 or G =G1 ∪G2 has an overall single edge, then

E[ψG1(i,µ)ψG2(j,ν)]−EψG1(i,µ)EψG2(j,ν) = 0 (11.51)

by independence between ψG1 and ψG2 . If G has no single edges, then by similar to argumentsof Lemmas 11.7 and 11.9, one may show that the number of noncoincident vertices of G is notmore than 2`.

11.4 e Ambainis-Harrow-Hastings ensemble and its general-ization

We turn now on the random tensor mixture ensembles. We will start with the (AHH)-scenario byquoting a result of [6]. We will use the same notation introduced in the previous sections.

eorem 11.10 (Ambainis, Harrow and Hastings, [6]). Let us consider the self-adjoint operator

Wn,c = ΨΨ† (11.52)

where the m columns of Ψ are vectors |ψµ〉 of the form

|ψµ〉 = |ψ1µ〉 ⊗ |ψ2

µ〉 ⊗ · · · ⊗ |ψkµ〉 , (11.53)

and the |ψξµ〉’s are i.i.d. with |ψξµ〉 ∼ Unif(H) (dimH = d). We denote n ≡ dk and we supposethat lim

n→∞

m

n= c ∈ ]0,+∞[. en, the empirical spectral measure of Wn,c weakly converges in

expectation to theMP(c) law

E%n,c %c , as n→∞ . (11.54)

In [6] the authors suggest the following

Conjecture 11.1 (Ambainis, Harrow and Hastings, [6]). e conclusion of the previous theoremremains true when convergence in expectation is replaced by almost sure convergence. at is,under the previous hypotheses

%n,c %c a.s. , as n→∞ . (11.55)

175

Remark 11.1. e two dierent notions of convergence are discussed in Section 1.3.1 (see alsoExample 1.2). We stress the fact that in [6] the authors do not consider the full spectrum ofWn,c butonly the R = min(m, dk) non-zero eigenvalues. Being normalized to 1, the limiting distributionof their paper diers from the absolutely continuous part of (11.14) of our eorem 11.1 above bya multiplicative factor c−1.

Our main result on the (AHH)-ensemble and its generalization (gAHH) is the followingtheorem.

eorem 11.11 (Random Tensors ensemble). Let consider the selfadjoint operator

Wn,c = ΨΨ† (11.56)

where the m columns of Ψ are vectors |ψµ〉 of the form

|ψµ〉 = |ψ1µ〉 ⊗ |ψ2

µ〉 ⊗ · · · ⊗ |ψkµ〉 , (11.57)

and the |ψξµ〉’s are i.i.d. with |ψξµ〉 ∼ Unif(H(ξ)) (dimH = dξ). We will considerWn,c as actingon an n-dimensional space with n ≡

∏ξ∈[k] dξ and we suppose that lim

n→∞

m

n= c ∈ ]0,+∞[. en,

the empirical spectral measure ofWn,c weakly converges almost surely to theMP(c) law.

e case h(ξ) ≡ H for all ξ corresponds to the (AHH)-ensemble and then eorem 11.11proves that Conjecture 11.1 is true. For saving notation, let us consider the case h(ξ) ≡ H for all ξ.It will be clear in the following that the generalized case could be proved in an almost verbatimway. Again, we apply the Gaussianization and the truncation tricks and we may assume that

Wn,c =1

nΨΨ† (11.58)

where n ≡ dk and

Ψ = (ΨIµ)I∈[d]k, µ∈[m] (11.59)

with ΨIµ := ψ(1)i1µψ

(2)i2µ· · ·ψ(k)

ikµ, (iξ ∈ [d] for all ξ ∈ [k]). e components ψ(ξ)

iξµcan be assumed to

be i.i.d. with|ψ(ξ)iξµ| < C , Eψ

(ξ)iξµ

= 0 , E|ψ(ξ)iξµ|2 = 1 , (11.60)

for all i ∈ [d], µ ∈ [m] and ξ ∈ [k]. In the language of multivariate statistics,Wn,c in (11.58) isa covariance matrix of a dataset (the Ψ in (11.59)) with nonindependent entries (see more laterin Section 11.9.1). For notation convenience we will write ψiξξ,µ ≡ ψ

(ξ)iξµ

. In this scenario, the

176

average traces ofWn,c are given by:1

nE[trW`

n,c

]=

1

dkE[trW`

n,c

]1 = d−k(`+1)E

[tr(

ΨΨ†)`]

= d−k(`+1)E∑

I1,...,I`∈[d]k

∑µ1,...,µ`∈[m]

ψI1µ1ψI2µ1ψI2µ2ψI3µ2 · · ·ψI`µ`ψI1µ`

= d−k(`+1)E∑

ψi11µ1ψi12µ1 · · ·ψi1kµ1ψi21µ1ψi22µ1 · · · ψi2kµ1

...ψi`1µ`ψi`2µ` · · ·ψi`kµ`ψi11µ`ψi12µ` · · · ψi1kµ`

= d−k(m+1)E∑

(ψi11µ1ψi21µ1ψi21µ2ψi31µ2 . . . ψi`1µ`ψi11µ`)

(ψi12µ1ψi22µ1ψi22µ2ψi32µ2 . . . ψi`2µ`ψi12µ`)

...(ψi1kµ1ψi2kµ1ψi2kµ2ψi3kµ2 . . . ψi`kµ`ψi1kµ`) . (11.61)

As usual, in order to apply the moment method we wish to identify a manageable set of diagrams.To deal with tensors we need the extension from bipartite graphsG(i,µ) (k = 1) of Section 11.3.3to (k + 1)-partite graphs G(i1, . . . , ik;µ) whose edges are now coloured with k distinct coloursξ’s. e “down” edges of colour ξ ∈ [k] of the graph are (ia,ξ, µa), a ∈ [`] and the “up” edgesof colour ξ are (µa, ia+1,ξ), a ∈ [`], with the convention that i`+1,ξ = i1,ξ for all ξ’s. A (k + 1)-partite graph can be obtained in a natural way by gluing k bipartite graphs G(i1, . . . , ik;µ) ≡G(i1,µ) ∪ G(i2,µ) ∪ · · · ∪ G(ik,µ). See Figure 11.3. Again we can partition the graphs inisomorphism classes represented by canonical ∆(k)(`, r1, . . . , rk, s)-graphs and we identify theseisomorphism classes ∆

(k)1 (`, s), ∆

(k)2 (`, r1, . . . , rk, s) and ∆

(k)3 (`, r1, . . . , rk, s) dened as before

in terms of the constituent bipartite graphs. For each m, r, s1, . . . , sk, the number of graphs inthe isomorphism class represented by the canonical ∆(`, r1, . . . , rk, s)-graph is

#∆(`, r1, . . . , rk, s) = m(m− 1) . . . (m− (s− 1))∏ξ∈[k]

d(d− 1) . . . (d− rξ)

= msd∑ξ rξ+1

[1 +O(d−1)

]. (11.62)

We will denote the generic word in (11.61) by

ψG(i1,...,ik;µ) ≡ ψG(i1,µ) . . . ψG(ik,µ) ≡∏ξ∈[k]

ψi1ξµ1ψi2ξµ1ψi2ξµ2ψi3ξµ2 · · ·ψi`ξµ`ψi1ξµ` . (11.63)

Recalling that dierent colours are statistically independent we have the following lemmas.

177

G(i1,µ) ∪G(i2,µ)=

i1

µ

i11 = i41

µ1 = µ3

i21 i31

µ2

i2

µ

⋃µ1 = µ3

i21

µ2

i12 = i42 i22 = i32

i11 = i41

µ1 = µ3

i21 i31

µ2

i12 = i42 i22 = i32

=G(i1, i2;µ)

i2

i1

µ

ψG(i1,i2;µ) = |ψi11µ1 |2ψi21µ1ψi21µ2ψi31µ2ψi31µ1 |ψi12µ1 |

2|ψi22µ1 |2|ψi22µ2 |

2

Figure 11.3: Examples of (k + 1)-partite graph. Here the number of dierent colours is k = 2. A (k + 1)-partitegraphs is obtained by gluing k bipartite graphs.

Lemma 11.12 (Rainbow diagrams). For ` and s xed, the number of ∆(k)1 (`, s)-graphs is

|∆(k)1 (`, s)| = |∆1(`, s)| = Nar(`, s) . (11.64)

Moreover Eψ∆

(k)1 (`,s)

= 1.

Lemma 11.13. For all ∆(k)2 (`, r1, . . . , rk, s): Eψ

∆(k)2 (`,r1,...,rk,s)

= 0.

Lemma 11.14. e total number of noncoincident vertices ks+∑

ξ(rξ+1) of a ∆(k)3 (`, r1, . . . , rk, s)-

graph satises the inequality ks+∑

ξ rξ < k`.

Proof of Lemmas 11.12, 11.13 and 11.14 . Most of the claims are a direct consequence of Lem-mas 11.7, 11.8 and 11.9 and the fact that the (k+ 1)-partite graph can be obtained in a natural wayby gluing k statistically independent bipartite graphs G(i1, . . . , ik;µ) ≡ G(i1,µ) ∪G(i2,µ) ∪· · · ∪ G(ik,µ) with a common µ-line. For instance one observe that ∆

(k)1 (`, s) is obtained by

gluing k bipartite canonical graphs ∆1(`, s) with the same µ-line. Since a ∆1(`, s)-graph iscompletely specied by `, s and µ, the (k + 1)-partite canonical graph ∆

(k)1 (`, s) should be a

178

i1

i1

i1

µ1

(1, 1)

(1, 2)

(1, 3)

(2, 1)

(2, 2)

(2, 3)

∆(3)1 (3, 1) = |ψ11,1|2|ψ21,1|2|ψ12,1|2|ψ22,1|2|ψ13,1|2|ψ23,1|2

Figure 11.4: Examples of ∆(k)1 -graph (rainbow diagram). Here the number of dierent colours is k = 3.

“rainbow diagram” (see Figure 11.4). en, the number of ∆(k)1 (`, s)-graphs concides with the

number of ∆1(`, s)-graphs. e proof of Lemma 11.13 is trivial. Regarding Lemma 11.14 it issucient to observe that by Lemma 11.9, for all ξ ∈ [k]: rξ + 1 + s ≤ `. Summing over ξ we getks+

∑ξ rξ ≤ k(`− 1) < k` and then the claim.

Proof of eorem 11.11. e technique is again the moment method. We will show that the averagespectral moments of Wn,c = n−1ΨΨ† converge to the moment of the MP(c) law and thevariances are summable. Similarly to the proof of Lemma 11.5 we can compute the averagespectral moments. Neglecting subleading terms, we get from (11.61)

1

nE[trW`

n,c

]= d−k(`+1)

∑i1,i2,...,ik;µ

EψG(i1,i2,...,ik;µ)

= d−k(`+1)∑

∆(k)(`,r1,...,rk,s))

msd∑ξ∈[k] rξ+1[1 +O(d−1)]Eψ∆(`,r1,...,rk,s)

179

=d−k(`+1)∑

∆(k)1 (`,s)

msdk(`−s+1)Eψ∆1(`,s)

+d−k(`+1)∑

∆(k)2 (`,r1,...,rk,s)

msd∑ξ∈[k] rξ+1Eψ∆2(`,s) (identically zero by Lemma 11.13)

+d−k(`+1)∑

∆(k)3 (`,r1,...,rk,s)

msd∑ξ∈[k] rξ+1Eψ∆3(`,s) (innitesimal by Lemma 11.14)

=∑

∆1(`,s)

csEψ∆1(`,s)

=∑s=1

cs|∆1(`, s)| →∑s=1

cs Nar(`, s) .

For the variance we have

Var

[1

ntrW`

n,c

]=

1

d2k

E[(trW`

n,c

)2]−E

[trW`

n,c

]2

= d−2k(`+1)∑i1,...,ikj1,...,jkµ,ν

E[ψG1(i1,...,ik;µ)ψG2(j1,...,jk;ν)]−EψG1(i1,...,ik;µ)EψG2(j1,...,jk;ν)

en, using the independence of the subgraphs of a (k + 1)-partite graph and a similar argumentas in the proof of Lemma 11.6, one can show that the number of summand that give a nonzerocontribution to the variance is bounded by d2k` and conclude that

Var

[1

ntrW`

n,c

]= O(n−2) . (11.65)

e case (gAHH) is proved in the same way. e only dierences are that for the (gAHH)-ensemble we have n =

∏ξ∈[k] dξ and the cardinality of the class represented by (`, r1, . . . , rk, s)

is ms∏ξ∈[k] d

rξ+1ξ [1 + o(1)]). It is worthy to point out the departures of our approach from the

strategy followed in [6] to prove 11.10. In [6] the authors present three dierent approaches tocompute the average moment. eir diagrammatic method proceeds as follows: among all thewords ψG(i1,...,ik;µ) they remove the ones with zero contribution (the ∆2-type graphs). en,they show that, among all the non-trivial words the leading contribution is given by the “rainbowdiagrams” words. On the contrary, our method relies on the standard diagrammatic for Wishartmatrices as presented in Section 11.3.3. Within this method the words are partitioned into threeclasses: words that give null contribution ∆2, rainbow diagrams ∆1 whose contribution we areable to compute explicitly, and the rest ∆3. e advantage of this procedure is that we do notneed to compute or estimate the contribution of a ∆3-graph (which is in general an hard task)

180

but it is sucient to show that the ∆3-graphs give a subleading contribution by controlling theircardinality. Moreover, this diagrammatic can be easily employed to estimate the variance of themoments and then to prove the almost sure convergence. Our problem has a deep connection withconvolution of free random variables. is connection has been intuited by Ambainis, Harrowand Hasting [6] who observed that the Narayana number involved in the diagrammatic approachcan be interpreted as numbers of non-crossing partitions of a certain set. As the convolution ofindependent random variables involves partitions, the free convolution of freely independentrandom variables involves non-crossing partitions. For a review on the combinatorial approachto free random variables see [204] and [166]. In the following Section we will show that freeprobability provides a neat and unied approach for mixture of random projections.

11.5 A Free Probability approach

Free addition of large random matrices is a noncommutative counterpart of addition of inde-pendent random variables in classical probability. Let An and Bn two matrices. It is knownthat, in general, the spectrum of An +Bn is not simply a function of the spectra of An and Bnindividually. In terms of empirical spectral densities

%An+Bn 6= f(%An , %Bn) . (11.66)

In general, more information is needed, namely the relative “orientation” of the eigenbases of Anand Bn. One can, however, dene a new type of addition of matrices in such a way that (11.66)becomes an equality (for some complicate but universal function f ). Indeed, this happens if weconsider the spectrum of

UAnU† + V BnV

† , (11.67)

where U, V are independent Haar unitaries (see Denition 2.6), for instance Haar-distributedunitary matrices. e transformation An 7→ UAnU

† and Bn 7→ V BnV† preserves the individual

spectra but “randomize” the eigenbases in such a way that the eigensystems of An and Bn are ina “generic position”. In such a case, for large dimensions n the rules of free additive convolutionhold, and one can predict the moments of UAnU † + V BnV

† from the individual moments of Anand Bn. For more details see [226].

e idea is that our operatorWn,c in Table 11.1 is essentially a sum of random projections in“generic position”. Is it possible to use the free additive convolution “” in order to compute thelarge size empirical spectral density ofWn,c?

11.6 Insights from Group eory

In this Section we will rephrase our original problem in a new language1. Hereaer U(d) willdenote the group of unitary operators on a d-dimensional space H ' Cd: U(d) ' U(H). Let us

1e title and the language of this Section is inspired from [48]

181

consider the self-adjoint operator

Wn,c =∑µ∈[m]

Π(gµ) |φ0〉 〈φ0|Π(gµ)† ∈ B(H ) . (11.68)

Here Π: G → U(H ) is a unitary representation of a compact group G: Π(g−1) = Π(g)−1 =Π(g)† and Π(g)Π(g−1) = Π(g−1)Π(g) = 1H . e ducial vector |φ0〉 6= 0 belongs to aninvariant subspace of the representation Π and the gµ are i.i.d. with respect to the G-invariantlaw.

e standard representation: (MP) scenario Let us consider the group G = U(d). isgroup is naturally represented by the standard representation

Π: U(d) 3 U 7→ Π(U) = U ∈ U(H) (11.69)

that provides an irreducible representation of U(d). e dimension of the representation is n = d.Moreover the group of unitary transformations U(H) acts transitively on the space of pure space(S(H) is an homogenous space). For every ducial vector |ψ0〉 ∈ S(h)

span Π(g) |φ0〉 : g ∈ U(d) = H . (11.70)

External product representation: (AHH) and (gAHH) scenarios Let us consider the groupdirect product G = U(d1)× · · · × U(dk) for k > 1. e standard representations Πξ : U(dξ) 3Uξ 7→ Πξ(Uξ) = Uξ ∈ U(H(ξ)) with dimH(ξ) = dξ are irreducible. en, the external productrepresentation

Π: U(d1)× · · · × U(dk) 3 U = (U1, . . . , Uk) 7→ Π(U) = U1 ⊗ · · · ⊗ Uk ∈ U(H ) , (11.71)

is an irreducible representation of G. e representation space is the tensor product H =⊗ξ∈[k] H

(ξ) of the individual representation spaces of U(dξ) and its dimension is n =∏ξ∈[k] dξ .

e unit sphere S(H ) is not an homogenous space for the group of local unitaries U(H(1))×· · ·×U(H(k)) (see discussion at the end of Section 8.1). Nevertheless, the external product representationhas no nontrivial invariant subspaces and then, for every ducial vector |ψ0〉 ∈ S(H ):

span

Π(U) |φ0〉 : U ∈ U(h(1))× · · · × U(h(k))

=⊗ξ∈[k]

H(ξ) . (11.72)

e tensor product representation: (F), (Sym) and (Anti) scenarios Let us dene the actionof Sk and U(d) on the tensor product space H = H⊗k where dimH = d. Let us x a basis|ei〉i∈[d] of H. en, the natural representation of the symmetric group Sk

V : Sk 3 π 7→ V (π) ∈ U(H ) (11.73)

182

is dened by

V (π) |ei1〉 ⊗ |ei2〉 ⊗ · · · ⊗ |eik〉 = |eπ−1(i1)〉 ⊗ |eπ−1(i2)〉 ⊗ · · · ⊗ |eπ−1(ik)〉 . (11.74)

e tensor product representation of G = U(d) in U(H ) is

Π: U(d) 3 U 7→ Π(U) = U⊗k ∈ U(H ) . (11.75)

Both V and Π are reducible representations of Sk and U(d), respectively. However, one canshow that Sk and U(d) are commutants [48] in the following sense. Let us denote A the algebragenerated by V (π) π∈Sk and C the algebra generated by Π(U) U∈U(d). Both A and C aresubalgebras of B(H ). en A′ = C and C′ = A, where for every subset K ∈ B(H ) itscommutant K′ is

K′ = X ∈ B(H ) : XK = KX for all K ∈ K . (11.76)

en, the full space H decomposes as the direct sum of invariant subspaces of Π and V :

H =⊕

λ`(k,d)

Πλ ⊗ Vλ . (11.77)

In (11.77), Πλ (resp. Vλ) denote the invariant subspaces of the reducible representation Π of(11.75) (resp. (11.73)). ey are labelled by Young frames with k boxes and no more than d rows.One of these sectors is the totally symmetric space

Hk =1

k!

∑π∈Sk

V (π)H⊗k . (11.78)

is invariant subspace has dimension n =(d+k−1k

). For every |ψ0〉 ∈ Hk

span Π(g) |φ0〉 : g ∈ U(d) = Hk . (11.79)

If k ≤ d, another invariant subspace of the tensor product representation Π is the totally anti-symmetric subspace

H∧k =1

k!

∑π∈Sk

σπV (π)H⊗k , (11.80)

where σπ denotes the parity of π. is subspace has dimension n =(dk

). In this case, for every

|ψ0〉 ∈ H∧k

span Π(g) |φ0〉 : g ∈ U(d) = H∧k . (11.81)

183

Wn,c =∑

µ ψµ Π(g) |φ0〉 n = dim Π

(MP) U ∈ U(H) |φ0〉 ∈ H d

(AHH) U1 ⊗ · · · ⊗ Uk, Uξ ∈ U(H) |φ0〉⊗k ∈ H⊗k dk

(gAHH) U1 ⊗ · · · ⊗ Uk, Uξ ∈ U(Hξ) |φ0〉⊗k ∈ H⊗k∏ξ∈[k] d

kξ

(F) U⊗k, U ∈ U(H) |φ0〉⊗k ∈ H⊗k(d+k−1k

)(Sym) U⊗k, U ∈ U(H) |φ0〉 ∈ Hk

(d+k−1k

)(Anti) U⊗k, U ∈ U(H) |φ0〉 ∈ H∧k

(dk

)(Perm) P ∈ Sd |φ0〉 ∈ H depends on |φ0〉

Table 11.2: Summary of the various ensembles of random tensors.

Standard representation of the Symmetric group: (Perm) ensemble Let us dene a uni-tary representation of Sd in H via

Π(π) |ψ〉 = Π(π)∑i∈[d]

ψi |ei〉 =∑i∈[d]

ψi |eπ(i)〉 . (11.82)

where |ei〉i∈[d] is a (xed) orthonormal basis of H. is representation should not be con-fused with what we called natural representation of the symmetric group (see (11.74)). Here therepresentation space is H with dimension n = d. We observe that

span Π(π) |φ0〉 : π ∈ Sd = H . (11.83)

as long as the ducial vector |φ0〉 =∑

i∈[d] φ0i |ei〉 has no repeated Fourier coecients (i.e.# φ0i , i ∈ [d] = d).

e following scheme (summarized in Table 11.2) will explain the connection with the ensem-bles introduced at the beginning of the Chapter.

11.7 A general problem and the Free Poisson law

A general problem in free probability (and then in RMT) is the following (see [165]).

Problem 11.1. For all n ≥ 1 let (Un,i)i∈I be a family of random unitaries (see Denition 2.6)distributed according to the Haar measure on the closed subgroup Gn of U(n). Is it true that fora reasonable sequence of subgroups (Gn)n≥1 the random variables (Un,i)i∈I are asymptoticallyfree for n→∞?

Hereaer we will denote, for every n ≥ 1, ϕn := 1nEtrn and φn := 1

ntrn, where the trace trnis on n-dimensional matrices. e answer to Problem 11.1 is positive in the caseGn ≡ U(n) for alln ≥ 1 (see eorem 2.9). So, roughly speaking, random unitaries Un,i independently distributed

184

according to the Haar measure on the unitary group U(n) are in a “generic position”. Nica showsin [165] that the answer is armative also in the case Gn ≡ Sn the subgroup of permutations.

Here we show that this “generic position” property is true also for local unitaries (eo-rems 11.15 and 11.16) and unitary transformations of indistinguishable systems (eorem 11.17).

eorem 11.15 (Asymptotic freeness of random local unitaries). Let n =∏j∈[k] dj with di

integers ≥ 2 and let Gn = U(d1) × U(d2) × · · · × U(dk), ϕn = E⊗

j∈[k]1dj

trdj . If for all n the

elements Un,i are pairwise independent and for every i ∈ I , Un,i = U(1)n,i ⊗U

(2)n,i ⊗ · · · ⊗U

(k)n,i with

the U (j)n,i ’s independently distributed according to the Haar measure on U(dj) with

∏ξ∈[k] dξ = n,

then the answer to Problem 11.1 is positive.

e theorem can be strengthened to include deterministic matrices as follows.

eorem 11.16. Let (Un,i)i∈I be a family of independent n × n unitary matrices with Un,i =⊗ξ∈[k] U

(ξ)n,i , each U (ξ)

n,i being independently distributed according to the Haar measure on U(dξ)

for all ξ ∈ [k] (and∏ξ∈[k] dξ = n). Let also

(Bn,t = B

(1)n,t ⊗ · · · ⊗B

(k)n,t

)t∈T

be a family of deter-

ministic n×nmatrix. en((Un,i, U

†n,i

)i∈I

, (Bn,t, B†n,t)t∈T

)is a family of pairs asymptotically

freely independent almost everywhere with respect to φn := 1ntrn. As a consequence the family is

also asymptotically free with respect to ϕn := 1nEtrn.

Proof. In the proof we will use the fact that random unitary matrices are Haar unitaries (as non-commutative random variables) so that we can limit ourselves to consider only their monomialsin proving the asymptotic freeness result. Our proof technique follows the technique of Hiai andPetz [108]. We may assume without loss of generality that the (Bn,t)t∈T,n∈N form a ∗-subalgebraof∏n∈NMn(C). Indeed we will prove the following: if i1, . . . , il ∈ I , m1, . . . ,ml ∈ Z − 0,

and t1, . . . , tl ∈ T are such that for each r ∈ [l] either

(i) trn(Bn,tr) = 0 (n ∈ N), or

(ii) Bn,tr = In (n ∈ N) and sr 6= sr+1 with sl+1 := s1,

then

trn

∏r∈[l]

Umrn,srBn,tr

→ 0 a.s. , asn→∞ . (11.84)

By eorem 2.1 of [108], for each factor j ∈ [k], if either condition i) or ii) is true, then

trdj

∏r∈[l]

U (j)n,sr

mrB

(j)n,tr

→ 0 a.s. , asn→∞ . (11.85)

e heart of the proof of (11.85) is combinatorial and we refer to [108] or [68] for details. econclusion (11.84) of the theorem is a trivial consequence of trn = trd1 ⊗ · · · trdk .

185

eorem 11.17 (Asymptotic freeness for k-fold random unitaries). e same conclusion of e-orem 11.16 holds if (Un,i)i∈I are independent n × n unitary matrices with Un,i = U⊗kn,d,i (withdk = n), each Un,d,i being independently distributed according to the Haar measure on U(d). Let

also(Bn,t = B

(1)n,t ⊗ · · · ⊗B

(k)n,t

)t∈T

be a family of deterministic n×n matrix. en the family of

pairs((Un,i, U

†n,i

)i∈I

, (Bn,t, B†n,t)t∈T

)is asymptotically freely independent almost everywhere

with respect to φn := 1ntrn. As a consequence the family is also asymptotically free with respect to

φn := 1nEtrn.

e Marcenko-Pastur law for mixture of random projection operators can be elegantly provedusing the tools of free probability. is problem is a noncommutative analogue of “balls-into-bins”problem in classical probability. By analogy with the classical Poisson law, the free Poisson law isthe limit distribution of a sum of identically freely independent variables[(

1− 1

n

)δ0 +

1

nδ1

]n. (11.86)

We report the details of the derivation of this important law.

eorem 11.18 (Free Poisson law). For all n ≥ 1 let µn be the measure

µn =

(1− 1

n

)δ0 +

1

nδ1 . (11.87)

If limn→∞

m

n= c ∈ (0,+∞), then

limn→∞

µmn =MP(c) . (11.88)

Proof. Although this is a standard result in free probability, we present here the proof in detailsfor the sake of completeness. e Stieltjes transform of µn (see (2.9)) is

sµn(z) =

ˆdµn(λ)

z − λ=

(1− 1

n

)1

z+

1

n

1

z − 1. (11.89)

e R-transform Rµn (see Denition 2.5) is the solution of

sµn(Rµn(z) + 1/z) = z . (11.90)

A straightforward computation gives

Rµn =1

(1− z)n+O(n−2) . (11.91)

Recall (eorem 2.8) that the R-transform behaves well under free convolution, i.e. for everymeasures µ and ν:

Rµν(z) = Rµ(z) +Rν(z) . (11.92)

186

enRµ := lim

n→∞Rµmn = lim

n→∞mRµn =

c

1− z. (11.93)

Now we have to nd the distibution µ with R-transform given by (11.93). Going backwards, wend rstly sµ using (11.90)

1

sµ(z)+Rµ(sµ(z)) = z , (11.94)

and then µ by the Stieltjes inversion formula (2.17). From (11.90) it is elementary to get

sµ(z) =1

2z

[c− 1− z +

√(c− 1− z)2 − 4z

], (11.95)

and the inversion formula (2.17) provides exactly the Marcenko-Pastur law with parameter c:

dµ(λ)

dλ= max 0, (1− c) δ(λ) +

1

2πλ

√(λ+ − λ)(λ− λ−) 1λ−≤λ≤λ+ (11.96)

11.8 Main results

Now we can state the main results of this Chapter. e proofs of these results are an immediateconsequence of the almost surely asymptotic freeness stated in eorems 11.15 and 11.16 and theFree Poisson law of eorem 11.18.

eorem 11.19 (Generalized Marcenko-Pastur eorem I). For k ≥ 1, let d1, . . . , dk ≥ 2 be somepositive integers and let n =

∏ξ∈[k] dξ . Let us consider the following self-adjoint operator acting on

the n-dimensional space H

Wn,c =∑µ∈[m]

(Uµ(1) ⊗ · · · ⊗ Uµ(k)) |ψ0〉 〈ψ0| (Uµ(1) ⊗ · · · ⊗ U (k)

µ )† ∈ B(H ) . (11.97)

Here Uµξ ∈ U(dξ) for all ξ ∈ [k] are independently distributed according to the Haar measureon the unitary group U(dξ) for all µ ∈ [m] and |ψ0〉 ∈ H . Suppose that m = m(n) withlimn→∞

m

n= c ∈ (0,+∞). en, the empirical spectral density %n,c of Wn,c weakly converges

almost surely to the Marcenko-Pastur lawMP(c).

eorem 11.20 (Generalized Marcenko-Pastur eorem II). For k > 1, let us consider the tensorproduct representation Π(U) = U⊗k of U(d) (d > 1). Let |ψ0〉 be a unit vector in the invariantsubspace H ≡ Πλ ⊂ H⊗k and suppose that dim Πλ →∞ as d→∞. Let us consider the followingself-adjoint operator acting on the n-dimensional space H

Wn,c =∑µ∈[m]

(Uµ⊗k) |ψ0〉 〈ψ0| (Uµ⊗k)† ∈ B(H ) . (11.98)

187

Here the Uµ ∈ U(d) are independently distributed according to the Haar measure on the unitarygroup U(d) for all µ ∈ [m]. Suppose that m = m(n) with lim

n→∞

m

n= c ∈ (0,+∞). en, the

empirical spectral density %n,c ofWn,c weakly converges almost surely to the Marcenko-Pastur lawMP(c).

Proof of eorems 11.19 and 11.20. Both theorems are an immediate consequence of the asymp-totically freeness almost everywhere of the family((

(Uµ(1) ⊗ · · · ⊗ Uµ(k)), (Uµ

(1) ⊗ · · · ⊗ Uµ(k))†)µ∈[m]

, |ψ0〉 〈ψ0|)

(11.99)

in eorem 11.19, and the family(((Uµ

⊗k), (Uµ⊗k)†

)µ∈[m]

, |ψ0〉 〈ψ0|)

(11.100)

in eorem 11.20. Hence the claim follows from the free Poisson law (eorem 11.18) where n isthe dimension of H .

eorem 11.21 (Generalized Marcenko-Pastur eorem III). Let us consider a n-dimensionalHilbert space H and let us x a basis ei i∈[n] of H . Let us consider the natural representationΠ(π), π ∈ Sn ofSn (n > 1) dened by (11.82). Let |ψ0〉 =

∑i∈[n] φ0i |ei〉with # φ0i , i ∈ [n] =

n be a unit vector in H . Let us consider the following self-adjoint operator acting on the n-dimensional space H

Wn,c =∑µ∈[m]

Π(πµ) |ψ0〉 〈ψ0|Π(πµ)† ∈ B(H ) . (11.101)

Here the Π(πµ) ∈ U(d) are independently distributed according to the Haar measure on the sym-metric group Sd for all µ ∈ [m]. Suppose that m = m(n) with lim

n→∞

m

n= c ∈ (0,+∞). en, the

empirical spectral density %n,c ofWn,c weakly converges almost surely to the Marcenko-Pastur lawMP(c).

Proof. In [165] it has been proved the asymptotically freeness almost everywhere for Haar-distributed random permutation. Hence the claim follows from the free Poisson law (eorem11.18).

e above eorems provide a complete understanding of the mixtures of random tensorsof Table 11.1. In particular, eorem 11.19 explains the (AHH)-ensemble and its generalization(gAHH) that we have already investigated using a diagrammatic approach. eorem 11.20 explainsin a neat way how the Marcenko-Pastur law emerge for the (F)-ensemble and their relatives(Sym) and (Anti). Finally, eorem 11.21 predicts that even the (Perm)-ensemble converges to theMP(c) law if the ducial vector is generic with respect to the natural action of the symmetricgroup. A numerical verication of these predictions is summarized in Fig. 11.5. In the Figure wecompare the Marcenko-Pastur law with a numerical estimation of the density. We also show theagreement of the moments with the theoretical prediction (11.12).

188

11.9 Final remarks

Besides solving a large class of random tensors problems, eorem 11.16 on the asymptoticfreeness of random local unitaries is per se valuable. We conclude this Chapter with a briefdiscussion on the implications of our result in Random Matrix eory.

11.9.1 Sample covariance matrices

We have seen that our mixtures of random tensors can be wrien as

W =1

nΨΨ† , (11.102)

where the entries of the n×m matrix Ψ are standardized but in general not independent. erandom matrixW is then a sample covariance-like matrix of the matrix Ψ of data. In this Sectionwe will stay in the Gaussian paradigm to enlighten the dierences with the standard scenario ofmultivariate statistics. We think that a concrete representation of the data matrix Ψ could help tounderstand the departures from the standard Wishart ensemble. For standard Wishart matrices(the classical Marcenko-Pastur seing) we consider as Ψ a d×m data matrix with independentstandard Gaussian entries

ΨMP =

ψ11 ψ12 . . . ψ1m

ψ21 ψ22 . . . ψ2m...

... . . . ...ψd1 ψd2 . . . ψdm

(11.103)

In such a case we have independent data ψiµ, (i ∈ [d], µ ∈ [m]) with E[ψiµ] = 0, E[|ψiµ|2] = 1.e (AHH)-ensemble (and its generalization) consists in considering dk ×m data matrices of theform (here we choose d = 2 and k = 2 for simplicity)

ΨAHH =

ψ

(1)11 ψ

(2)11 ψ

(1)12 ψ

(2)12 . . . ψ

(1)1mψ

(2)1m

ψ(1)11 ψ

(2)21 ψ

(1)12 ψ

(2)22 . . . ψ

(1)1mψ

(2)2m

ψ(1)21 ψ

(2)11 ψ

(1)22 ψ

(2)12 . . . ψ

(1)2mψ

(2)1m

ψ(1)21 ψ

(2)21 ψ

(1)22 ψ

(2)22 . . . ψ

(1)1mψ

(2)2m

. (11.104)

In this matrix, the ψξiξµ (ξ ∈ [k], iξ ∈ [d], µ ∈ [m]) are independent standard Gaussian. It is easyto verify that the matrix entries ΨIµ = ψ

(1)i1µ· · ·ψ(k)

ikµsatisfy

E[ΨIµ] = 0 E[|ψIµ|2] = 1 E[ψIµψ?Jν ] = 0 , (11.105)

i.e. they are standardized, uncorrelated but not independent. e (F)-ensemble arises from a datadatrix of the form

ΨF =

ψ11ψ11 ψ12ψ12 . . . ψ1mψ1m

ψ11ψ21 ψ12ψ22 . . . ψ1mψ2m

ψ21ψ11 ψ22ψ12 . . . ψ2mψ1m

ψ21ψ21 ψ22ψ22 . . . ψ1mψ2m

, (11.106)

189

with the ψiµ independent (again we have chosen (d = 2 and k = 2 for simplicity). It is easy toverify that the relations (11.105) holds for the matrix entries (again the entries are uncorrelated).Finally the data matrix associated to the (Perm)-ensemble is

ΨPerm =

ψ1 ψσ1(1) . . . ψσm−1(1)

ψ2 ψσ1(2) . . . ψσm−1(2)...

... . . . ...ψd ψσ1(d) . . . ψσm−1(d)

(11.107)

where σ1, . . . , σm−1 are independent uniformly distributed permutations and ψ1, . . . , ψd areindependent standardized Gaussians.

To anyone of these data matrices ΨMP, ΨAHH, ΨF and ΨPerm it is associated a covariancematrixW whose limiting law (with a prescribed ratio c) is the Marcenko-Pastur lawMP(c).

11.9.2 More generic positions

e intuition that unitarily invariance should give rise to “generic” orientations was proven byVoiculescu (eorem 2.10). He proved that if An and Bn are independent and have almost surelylimiting distributions, then An and UnBnU †n (Un independently distributed according to the Haarmeasure on the unitary group) are almost surely asymptotically freely independent. In such acase it is possible to compute the limit distribution of

An + UnBnU†n (11.108)

using the free analysis and we write symbolically

%An+UnBnU

†n

= %An %Bn , as n→∞ . (11.109)

eorems 11.19 and 11.20 tell that there are other ways to realize generic situations. Indeed, let usagain consider An and Bn independent with almost surely limiting distributions. Let us considera bipartition of Cn = Cn1 ⊗ Cn2 . If (Un1 , Un2) ∈ U(n1) × U(n2) is distributed according tothe Haar measure, then An and (Un1 ⊗ Un2)Bn(Un1 ⊗ Un2)† are almost surely asymptoticallyfreely independent. e generalization to partitions of the form Cn = Cn1 ⊗ Cn2 ⊗ · · · ⊗ Cnk(k > 2) is straightforward. It is also easy to see that asymptotic freeness holds also betweenAn and (Un1 ⊗ Un2)Bn(Un1 ⊗ Un2)† in the case that Un1 are deterministic matrices, Un2 areHaar-distributed unitaries and n2 goes o to innity with n. As a consequence, the free additioncan be simulated for large n also by these kinds of random sums

An + (Un1 ⊗ Un2)Bn(Un1 ⊗ Un2)† (11.110)An + (1n1 ⊗ Un2)Bn(1n1 ⊗ Un2)† . (11.111)

In both cases, the empirical spectral density of the sum converges to the free convolution

%An+(Un1⊗Un2 )Bn(Un1⊗Un2 )† = %An %Bn , as n→∞ , (11.112)

%An+(1n1⊗Un2 )Bn(1n1⊗Un2 )† = %An %Bn , as n→∞ . (11.113)

We provide an example.

190

Example 11.1. A quite surprising phenomenon in free probability is the following. Let a andb be random variables with equal measure µ = 1

2(δ0 + δ1). In classical probability, supposinga and b independent, the probability law of a + b is a discrete measure given by the classicalconvolution µa?b = µ ? µ = 1

4 (δ0 + 2δ1 + δ2). On the contrary the free convolution of twodiscrete probability distributions can be continuous. For instance, in our example, if a and b arefreely independent, it is easy to compute their free convolution

µab = µ µ =1

π√x(2− x)

10<x<2 . (11.114)

In matrix terms, the limiting distribution of the sum Pn + UnPnU†n

diag (1, 0, 1, 0, 1, . . . ) + Udiag (1, 0, 1, 0, 1, . . . )U † (11.115)

of two randomly rotated projection matrices of rank n/2 is given by the arcsine law (11.114).What we discovered is that the same limit distribution arises if we restrict ourselves to randomlocal randomizations of the form

diag (1, 0, 1, 0, 1, . . . ) + (U1 ⊗ U2)diag (1, 0, 1, 0, 1, . . . ) (U1 ⊗ U2)† , (11.116)

or even

diag (1, 0, 1, 0, 1, . . . ) + (1⊗ U)diag (1, 0, 1, 0, 1, . . . ) (1⊗ U)† . (11.117)

ese results are summarized in Fig. 11.6.

HgAHHL d1 = 3d2 = 5d3 = 8k = 3c = 4

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Figure 11.5: Comparison between the Marcenko-Pastur law (continuous line) and numerical simulations (dots) forthe generalized Amabainis-Harrow-Hasting (gAHH), the Tensor Fold (F) and Permutation ensemble (Perm). We alsocompare the moments E[trW`] of the Marcenko-Pastur law (empty black circles) and the numerical simulations (redpoints) for the three ensembles. e number of random summands (the rank-1 projections |ψµ〉’s is m = cn withc = 4. e sample size is N = 104. e error in the numerical simulation of the moments (red points) is not visible.

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Pn+U PnU Ö :

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Figure 11.6: Limit spectral distribution of the sum of two rank-n/2 projections in generic position. e generic positionis implemented in three dierent way as explained in Example 11.1. e dashed line is the arcsine distribution (11.114).e dots come from numerical simulation. Here the size n of the projection Pn is n = 625. e bipartition for localrandom unitaries U1 ⊗ U2 and 1⊗ U is balanced 25× 25. e sample size is N = 103.

Epilogue

Randomization in the study of large dimensional system is a well-established tool in manybranches of science. ese randomization ideas have been rened in time to the high level ofsophistication oered by random matrices.

is thesis is about the spectral methodologies for large dimensional random matrices andtheir applications. e thesis is tripartite.

In the rst Part, I detailed some properties of the classical ensembles. I decided to presentthe subject with a pedagogical avour and with a “rst examples” methodology (Chapter 1).Chapters 2 and 3 contain the basic results and techniques developed in RMT. Due to the limitationof my knowledge and space I restricted myself to those I beer understood and I used sincenow. Hopefully, the (many) others methodologies of RMT could be exploited to rediscover andstrengthen the results of Part II and III.

Part II was inspired by the way Large Deviation eory has been recently applied in order toestablish large deviation bounds for Coulomb gas system and unitarily invariant random matrixensembles. Most of the interesting phenomena for unitarily invariant ensembles are ultimatelyascribable to the logarithmic interaction of the eigenvalues (the level repulsion). One of the mostaractive features of the 2D Coulomb gas picture is that it gives an intuitive way to understandand guess some spectral phenomena of the classical ensemble. Moreover, 2D Coulomb gases areone of the exceptional solvable interacting systems, in the sense that, oen its partition functioncan be computed explicitly. In this thesis I presented a neat way to establish large deviationprinciples for linear statistics on random matrices and I provided an approach for the computationof their large deviation functions (Chapter 4). is approach is “new” from a methodologicalpoint of view in the sense that diers from the standard route used in the previous literature. isshortened route is computationally more ecient than the state-of-the-art. e main advantageof this method is the possibility to establish and compute the joint large deviation function fortwo or more linear statistics, a task almost unexplored since now in the literature.

I selected a few applications of the results contained in Chapter 4 in the context of multivariatestatistics (Chapter 5) and quantum transport in chaotic cavities (Chapter 6).

e second main result presented in Part II is a new universal covariance formula (Chapter 7).For a suciently smooth linear statistics An, there exist two celebrated variance formulas in thephysics literature by Dyson-Mehta (DM) and Beenakker (B). I set for myself the task to nd a

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universal formula for the covariance of two linear statistics An and Bn that would reduce toDM or B for An ≡ Bn. But before proceeding, it felt natural to rst check under which preciseconditions should we expect to recover one formula or the other. Much to my surprise, I did notnd a suciently transparent answer that encompasses all possible cases. In this thesis I thenintroduced a “conformal map” method which encloses all possible cases (old and new) into a neatand unied framework. Chapter 7 contains a thorough derivation of this new covariance formulaand a few applications as corollaries.

e last Part of the thesis was devoted to the application of random matrices in antumMechanics and antum Information. I have started this part by collecting some known results(Chapter 8). However, I point out that Chapter 8 also contains some new resulst. In fact, Itacitly applied the machinery developed in Part II to derive the joint behavior of the Renyi’sentropies, a result new to my knowledge. Moreover, I showed explicitly the joint law of theSchmidt eigenvalues of a random state constrained to a submanifold of xed degeneracy of theSchmidt eigenvalues. I did not speculate on this result but I think it would be interesting to dothat. For a purely RMT perspective, this result shows how to enhance the level repulsion in theclassical ensembles.

In Chapters 9, 10 and 11, I moved beyond the classical ensembles of RMT. Chapter 9 containsthe proposal of a new class of polarized ensemble. We introduced them as an operational pro-cedure to mimic ensembles of random pure states with a prescribed amount of entanglement.In fact these ensembles can be labeled by their typical purity and, in Chapter 10, we appliedthe spectral methodologies of RMT to investigate their spectral properties. We proved someempirical evidences and we hope that this seminal investigations could trigger the interest ofthe scientic community. Finally, we faced the problem of random tensors (Chapter 11). In thislast Chapter, I discussed in a systematic way the spectrum of mixture of random tensors. Wegeneralized the bipartite diagrammatic of random covariance matrices to these ensemble provinga recent conjecture. We also presented another method based on free probabilistic ideas. isnew approach justies, with the idea of “freeness”, many intuitive guesses and explains in a“conceptual way” the ubiquity of the Marcenko-Pastur law (macroscopic universality). In thisperspective, a main result of this Chapter is the proof of asymptotic freeness almost everywherefor random local unitaries. is result leaves a lot of room for speculations and applications.

ese conclusions ends with a separate discussion of open problems pertaining to Part II andIII.

Part II: Open problems

RMT is a very active eld in continue evolution. For convenience I collect here some openproblems. is list is not the List of long-standing problem in RMT. It is simply a collection ofproblems inspired by my studies that I nd worthy of aention.

ird-order phase transitions e large deviation function of a linear statisticsAn =∑

i a(λi)may exhibit non-analyticities in the large dimensional limit. ese singularities of the rate func-

195

tion are a manifestation of a phase transition in the underlying 2D Coulomb gas system. is is arecurrent phenomenon in RMT, and the list of evidences and solved models is surprisingly long. Atypical mechanism is the following. Suppose that the density of states of the Coulomb gas varieswith a control parameter s (the Laplace variable in the case of linear statics on random matrices)and that for all values of s the density of states is supported on a single interval and is constrainedto stay on a proper interval of the real line. In the physical literature this constraint is referredas the presence of a hard wall (in our presentation, the external potential V is innite outside aproper interval). Suppose that for s < sc the density vanishes as a square-root at one edge andthere is a nonzero gap between this edge and the hard wall. As s increases towards the criticalvalue sc this gap tends to zero. For s > sc the gap between the edge of the density of charges andthe hard wall vanishes and the density of states acquires a square-root divergence (the gas getspushed by the hard wall). Whenever this happens, the third derivative (with respect to s) of freeenergy of the gas is discontinuous at s = sc. If s is the Laplace variables of a linear statistics, thiscorresponds to a third-order non-analyticity of the large deviation functions. is phenomenon issurprisingly ubiquitous not only in RMT but in all the problems that admit a Coulomb gas picture.In every model that exhibits this hard wall mechanism the explicit computations reveal that theCoulomb gas experiences a third-order phase transition. Is it possible to transform this collectionof evidences and computations into a theorem? Such a theorem would be very helpful. Indeed,usually we can have a good understanding of the density of states of the Coulomb gas. However,without a theorem, we cannot predict that the hard wall mechanism (easy to recognize) implies athird-order phase transition unless we are able to compute explicitly the free energy (which isnot always an easy task!).

Comparison between noninteracting gases and log-gases Computing the partition func-tion Z of a mechanical system and then derive its thermodynamical properties is a formidabletask in general. Free particle systems and 2D Coulomb gases are in this sense quite exceptional:for these systems usually it is possible to compute Z! In Chapter 5 we made a digression tocompare the behavior of the noninteracting gases and log-gases. Under some conditions, it ispossible to compute (in the thermodynamical limit) average and variance of a sum functionA = n−1

∑i a(λi), where n is the number of particles of the system and λi denotes the position

of the ith particle. We pointed out some remarkable departures:

Average e average in the noninteracting case Efree[A] ∝ β−1 grows linearly with the temper-ature. For a log-gas, on the contrary, this average value is independent of the temperatureEint[A] ∝ β0.

Fluctuations e uctuations of A in the noninteracting case is√

Varfree(A) ∝ β−1 and√Varfree(A) = O(n−1/2). e scaling with n is the one predicted by the classical central

limit theorem and the uctuations around the average values increase linearly with thetemperature. Log-gases exhibit a completely dierent behavior:

√Varint(A) ∝ β−1/2

and√Varint(A) = O(n−1). e large n scaling corresponds to a stronger concentration

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of measure phenomena about the average value. Also remarkable is that the uctuationsgrows slower with the temperature compared with the noninteracting case.

ese evidences have been conrmed in other investigations that I did not report here. ecovariance formula presented in Chapter 7 provides a mathematical understanding of the peculiarbehavior of log-gases. However, a “conceptual” understanding of the departure of log-gases fromthe noninteracting case is very much called for. My naive understanding of the concentration ofmeasure phenomena for interacting systems is the following. In a system with a large numberof independent individuals (thinking to a social system could help), the central limit theorempredicts a normal behavior. e intuition is that the independence among the individuals makesextreme behaviors highly improbable. e larger the population the more suppressed are theuctuations around the typical behavior. In an interacting system (people communicate and sharefeelings and opinions in our social system), again the behavior of people is self-averaging (peoplebehave typically) but the interaction among the individuals enhance the suppression of abnormalfeatures (in a society, oen “non normal” is synonymous of “asocial”). Is it possible to establish amathematical framework of this intuition and apply it to physical systems?

A way to cure the non-steepness of the GF In Chapter 5 we faced the following problem.e cumulant generating function is dened on a proper interval of the real line and the steepnesscondition is not fullled. In such a case the Gartner-Ellis theorem is no longer true (even in itslocal version) and, in general, the Legendre transform of the cumulant GF J(s) is the convexenvelope of the rate function Ψ(x). ese situations have been already faced in the literature.However, as far as I know, the “solution” (whenever found) proposed in the physical literature isextremely “case-dependent” and oen it is not completely satisfactory. It would be desirable tohave a general way to cure such situations.

Multi-cut random matrix models e reader will have noticed that we almost always restrictour aention to one-cut matrix models, i.e. ensemble of invariant random matrices whose densityof states is supported on a single interval. Results on the one-cut models abound in the literature,but are much scarcer for multi-cut matrix models. Why should we care about them? I provide asimple but (hopefully) clear motivation. In this thesis I tackled the problem of establish a largedeviation principle for linear statistics. A fundamental hypothesis is that, in a neighborhood ofs = 0 (s being the Laplace parameter associated to the linear statistics) the saddle-point density%s = arg min Es[µ] has connected support. is is the underlying hypothesis we also used toderive the covariance formula for linear statistics. Essentially we always restrict ourselves tosomehow generically “smooth” linear statistics. A typical linear statistics that escapes fromthis framework is An =

∑i θ(x− λi), i.e. the number of eigenvalues smaller than a threshold

x ∈ R. In such a case, the linear statistics is not “smooth” and, indeed, for an arbitrary smallperturbation of the Laplace parameter s the support of the density of states is not connected.In a measure theory language, the mapM(R) 3 µ 7→ A[µ] =

´ xdµ(λ) is not continuos and

then, the contraction principle inevitably fails. In the derivation of the covariance formula, thefunctional derivative method also fails. Indeed, a small variation of the external potential causes

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an extreme change of the saddle-point density. From a physical point of view this is the veryobstruction that imposed to exclude the case of non-smooth observables.

General invariant models A lemma due to Weyl, states that any invariant ensemble law(Mn) =

law(UnMnU†n) is specied by the joint law of its rst rst traces

law(Mn) = law(trMn, trM2n, . . . , trM

nn ) . (1)

e joint law of the eigenvalues follows immediately as

P(λ) ∝ |∆(λ)|β f(∑i

λi,∑i

λ2i , . . . ,

∑i

λni ) , (2)

where the Vandermonde determinant pops out as a Jacobian when passing form the canonical(Mij) to the polar (λk, Uij) coordinates. So far we consider (less general) invariant matrix modelsof the form

law(Mn) ∝ e−trV (Mn)dMn . (3)

In such a case, (2) reduces to

P(λ) ∝ |∆(λ)|βe−∑k V (λk) , (4)

and we have seen how the Coulomb gas technology fruitfully applies to such a model. How-ever, a natural question is to develop such a technology for generic joint laws of the form (2).ese probability laws are not exotic at all. A trivial example is provided by a random ma-trix Mn = UDU † with U distributed according to the Haar measure on the unitaries andD = diag(λ1, . . . , λn) with the λi’s independently distributed. Such an ensemble is unitarilyinvariant by construction and, obviously, law(λ) =

∏k law(λk). Looking at (2), in some way, the

factor f(∑

i λi,∑

i λ2i , . . . ,

∑i λ

ni ) should cancel the Vandermonde determinant and provide a

factorized joint probability law (an explicit computation can be found in [224]). In the same spirit,more general invariant ensembles of the form (1) could generate non-logarithmic interactions,or three-particle (or more) interaction between the eigenvalues. A general overview of thesescenarios is still lacking in the literature.

Part III: Open problems

Here I propose a few open problems related to the application of RMT in antum Informationeory.

Connection of the unbiased ensemble and the Marcenko-Pastur ensemble In Section 8.3we have seen that, if |ψ〉 ∼ Unif(HA ⊗HB) the reduced state ωA = trB |ψ〉 〈ψ| is unitarilyinvariant. Moreover, if dB/dA → c ≥ 1, then the ESd %cdAωA weakly converges almost surely tothe Marcenko-Pastur law

%dBωA MP(c) . (5)

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On the other hand, eorem 11.2 says that the same macroscopic limit holds for the rescaledmeasure of the random mixture σA = d−1

B

∑µ∈[dB ] |ψµ〉 〈ψµ| with |ψµ〉 ∼ Unif(HA), i.i.d. (and

dB/dA → c ≥ 1)%dBσA MP(c) . (6)

By constructionσA is unitarily invariant law(σA) = law(UσAU†). By a theorem of Schrodinger [194]

(see also [167, 168]) it is known that ωA can be wrien as an equally weighted combination ofrank-1 projections:

ωA =1

dA

∑i∈[dA]

|ϕi〉 〈ϕi| , for some |ϕi〉’s . (7)

SinceωA is unitarily invariant, |ϕi〉 ∼ Unif(HA) for all i ∈ [dA] but the, in general, the |ϕi〉 cannotbe independent! Indeed, if they were so, by the Marcenko-Pastur theorem 11.2 %dBωA 9MP(c).What is the joint law of (|ϕ1〉 , . . . , |ϕn〉)?

Entropies of the polarized ensembles A study the full probability law of the purity and,more generally, of the Renyi’s entropies of the polarized ensemble would be important. It wouldbe also of great interest to treat the polarized ensemble within the abstract framework of the freesubordination developed in free probability theory. Hopefully, this approach could also revealsome information about the eigenvectors associated to the outlier(s) of the spectrum.

Volume of LOCC convertible states By a theorem of Schrodinger [194] (rediscovered severaltimes, see for instance [167, 168]) a pure state |ψ〉 ∈ HA ⊗HB can be converted into |φ〉 ∈HA ⊗HB using only local operations and classical communications (LOCC, see [47]) if and onlyif

λ(φ) ≺ λ(ψ) , (8)

where λ(ψ) (resp. λ(φ)) are the Schmidt eigenvalues of |ψ〉 (resp. |φ〉), and “≺” stands for themajorization relation for probability vectors (see [167] for instance). e problem is compute (orestimate)

Pr(|ψ〉 LOCC−−−→ |φ〉

), (9)

when |ψ〉 and |φ〉 are independent random pure states with |ψ〉 , |φ〉 ∼ Unif(HA⊗HB). Nielsen[167] conjectured that Pr

(|ψ〉 LOCC−−−→ |φ〉

)→ 0 as dA, dB → ∞. In Chapter 8 we showed that

the family of Renyi’s entropies is jointly asymptotically Gaussian. Since the LOCC-convertibilityimplies the majorization of the whole family of Renyi’s entropies, maybe our results can be usedto prove Nielsen’s conjecture.

Asymptotic freeness for the Weyl-Heisenberg group Let us consider Problem 11.1 forGn = Hn the Weyl-Heisenberg group. Is the answer armative for Hn? If yes, it should bepossible to establish a free Poisson law for random projections of the form Π(g) |ψ0〉 〈ψ0|Π(g)†

with g ∼ Unif(Hn).

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