Random Matrices

REVISED AND ENLARGED SECOND EDITION

MADAN LAL MEHTA Centre d'Etudes Nucleaires de Saclay

Gif-sur-Yvette Cedex France

Centre National de Recherche Scientifique France

Academic Press San Diego New York Boston

London Sydney Tokyo Toronto

Contents

Preface to the Second Edition xi Acknowledgments xv Preface to the First Edition xvii

1 / Introduction

1.1. Random Matrices in Nuclear Physics 1 1.2. Random Matrices in Other Branches of Knowledge 6 1.3. A Summary of Statistical Facts about Nuclear Energy Levels 10 1.4. Definition of a Suitable Function for the Study of Level Correlations 14 1.5. Wigner Surmise 15 1.6. Electromagnetic Properties of Small Metallic Particles 18 1.7. Analysis of Experimental Nuclear Levels 20 1.8. The Zeros of the Riemann Zeta Function 21 1.9. Things Worth Consideration, but Not Treated in This Book 33

2 / Gaussian Ensembles. The Joint Probability Density Function for the Matrix Elements

2.1. Preliminaries 36 2.2. Time-Reversal Invariance 37 2.3. Gaussian Orthogonal Ensemble 39 2.4. Gaussian Symplectic Ensemble 41 2.5. Gaussian Unitary Ensemble 46 2.6. Joint Probability Density Function for Matrix Elements 47 2.7. Another Gaussian Ensemble of Hermitian Matrices 52 2.8. Antisymmetric Hermitian Matrices 53 Summary of Chapter 2 53

3 / Gaussian Ensembles. The Joint Probability Density

Function for the Eigenvalues

3.1. Orthogonal Ensemble 55

v

vi Contents

3.2. Symplectic Ensemble 59 3.3. Unitary Ensemble 62 3.4. Ensemble of Antisymmetric Hermitian Matrices 65 3.5. Another Gaussian Ensemble of Hermitian Matrices 66 3.6. Random Matrices and Information Theory 67 Summary of Chapter 3 69

4 / Gaussian Ensembles. Level Density

4.1. The Partition Function 70 4.2. The Asymptotic Formula for the Level Density. Gaussian Ensembles 72 4.3. The Asymptotic Formula for the Level Density. Other Ensembles 75 Summary of Chapter 4 78

5 / Gaussian Unitary Ensemble

5.1. Generalities 80 5.2. The n-Point Correlation Function 89 5.3. Level Spacings 95 5.4. Several Consecutive Spacings 101 5.5. Some Remarks 109 Summary of Chapter 5 121

6 / Gaussian Orthogonal Ensemble

6.1. Generalities 123 6.2. Quaternion Matrices 125 6.3. The Probability Density Function as a Quaternion Determinant 128 6.4. The Correlation and Cluster Functions 135 6.5. Level Spacings. Integration over Alternate Variables 138 6.6. Several Consecutive Spacings: n = 2r 142 6.7. Several Consecutive Spacings: n = 1r — 1 147 6.8. Bounds for the Distribution Function of the Spacings 152 Summary of Chapter 6 160

7 / Gaussian Symplectic Ensemble

7.1. A Quaternion Determinant 162 7.2. Correlation and Cluster Functions 165 7.3. Level Spacings 167 Summary of Chapter 7 169

8 / Gaussian Ensembles: Brownian Motion Model

8.1. Stationary Ensembles 170

Contents VÜ

8.2. Nonstationary Ensembles 170 8.3. Some Ensemble Averages 176 Summary of Chapter 8 179

9 / Circular Ensembles

9.1. The Orthogonal Ensemble 182 9.2. Symplectic Ensemble 185 9.3. Unitary Ensemble 187 9.4. The Joint Probability Density Function for the Eigenvalues 188 Summary of Chapter 9 193

10 / Circular Ensembles (Continued)

10.1. Unitary Ensemble. Correlation and Cluster Functions 194 10.2. Unitary Ensemble. Level Spacings 197 10.3. Orthogonal Ensemble. Correlation and Cluster Functions 199 10.4. Orthogonal Ensemble. Level Spacings 206 10.5. Symplectic Ensemble. Correlation and Cluster Functions 210 10.6. Relation between Orthogonal and Symplectic Ensembles 212 10.7. Symplectic Ensemble. Level Spacings 214 10.8. Brownian Motion Model 216 10.9. Wigner's Method for the Orthogonal Circular Ensemble 218 Summary of Chapter 10 222

11 / Circular Ensembles. Thermodynamics

11.1. The Partition Function 224 11.2. Thermodynamic Quantities 227 11.3. Statistical Interpretation of U and C 230 11.4. Continuum Model for the Spacing Distribution 232 Summary of Chapter 11 238

12 / Asymptotic Behavior of Eß (0, s) for Large s

12.1. Asymptotics of the A„ (t) 240 12.2. Asymptotics of Toeplitz Determinants 243 12.3. Fredholm Determinants and the Inverse Scattering Theory 244 12.4. Application of the Gel'fand-Levitan Method 247 12.5. Application of the Marchenko Method 252 12.6. Asymptotic Expansions 255 Summary of Chapter 12 258

VÜi Conten t s

13 / Gaussian Ensemble of Antisymmetric Hermitian Matrices

13.1. Level Density. Correlation Functions 260 13.2. Level Spacings 263 Summary of Chapter 13 266

14 / Another Gaussian Ensemble of Hermitian Matrices

14.1. Summary of Results. Matrix Ensembles from GOE to GUE and Beyond . . 268 14.2. Matrix Ensembles from GSE to GUE and Beyond 275 14.3. Joint Probability Density for the Eigenvalues 279 14.4. Correlation and Cluster Functions 290 Summary of Chapter 14 293

15 / Matrices with Gaussian Element Densities but with No Unitary or Hermitian Conditions Imposed

15.1. Complex Matrices 294 15.2. Quaternion Matrices 301 15.3. Real Matrices 309 Summary of Chapter 15 310

16 / Statistical Analysis of a Level Sequence

16.1. Linear Statistics or the Number Variance 314 16.2. Least Square Statistic 319 16.3. Energy Statistic 324 16.4. Covariance of Two Consecutive Spacings 327 16.5. The F Statistic 330 16.6. The A Statistic 332 16.7. Statistics Involving Three- and Four-Level Correlations 333 16.8. Other Statistics 334 Summary of Chapter 16 338

17/Selberg's Integral and Its Consequences

17.1. Selberg's Integral 339 17.2. Selberg's Proof of Equation (17.1.3) 340 17.3. Aomoto's Proof of Equation (17.1.4) 345 17.4. Other Averages 349 17.5. Other Forms of Selberg's Integral 349 17.6. Some Consequences of Selberg's Integral 352 17.7. Normalization Constant for the Circular Ensembles 356

Contents IX

17.8. Averages with Laguerre or Hermite Weights 356 17.9. Connection with Finite Reflection Groups 359 17.10. A Second Generalization of the Beta Integral 361 17.11. Some Related Difficult Integrals 364 Summary of Chapter 17 369

18 / Gaussian Ensembles. Level Densi ty in t h e Tail of t he Semicircle

18.1. Level Density near the Inflection Point 372 Summary of Chapter 18 376

19 / Rest r ic ted Trace Ensembles. Ensembles Rela ted to t h e Classical Orthogonal Polynomials

19.1. Fixed Trace Ensemble 377 19.2. Bounded Trace Ensemble 381 19.3. Matrix Ensembles and Classical Orthogonal Polynomials 382 Summary of Chapter 19 384

20 / Bordered Matr ices

20.1. Random Linear Chain 387 20.2. Bordered Matrices 391 Summary of Chapter 20 393

21 / Invariance Hypothes is and Ma t r i x Element Correla t ions

21.1 Random Orthonormal Vectors 395 Summary of Chapter 21 399

Appendices

AI. Numerical Evidence in Favor of Conjectures 1.2.1 and 1.2.2 400 A2. The Probability Density of the Spacings Resulting from a Random

Superposition of n Unrelated Sequences of Enegy Levels 402 A3. Some Properties of Hermitian, Unitary, Symmetrie, or Self-Dual Matrices . . 405 A4. Counting the Dimensions of Tßc and TßG (Chapter 3) and of Tßc and TßC

(Chapter 8) 406 A5. An Integral over the Unitary Group, Equation (3.5.1) or Equation (14.3.1) . . 407 A6. The Minimum Value of W. Proof of Equation (4.1.6) 412

X Contents

A7. Relation between R„, T„, and E(n;26). Equivalence of Equations (5.1.3) and (5.1.4) 414

A8. Relation between E(n;s), F(n\s), and p(n;s): Equations (5.1.16), (5.1.17), and (5.1.18) 418

A9. The Limit of Y^!'1 <fj(x). Equation (5.2.17) 419 A10. The Limits of y]n <Pj(x)<Pj(.y), e t c- Sections 5.2, 6.4, and 6.5 421 A l l . The Fourier Transform of the Two-Point Cluster Functions 423 A12. Some Applications of Gram's Formula 426 A13. Power Series Expansions of Eigenvalues, of Spheroidal Functions, and of

Various Probabilities 428 A14. Numerical Tables ofAj(s), bj(s), and EB(n;s) for ß = 1, 2, and 4 432 A15. Numerical Values of Eß(0; s), <5lß(s), and pß (0, s) for ß = 1 and 2 and s < 3.7 437 A16. Proofof Equations (5.5.12)-(5-5.14) and (10.4.20) 438 A17. Correlation Functions the Hard Way 445 A18. Use of Pfaffians in Some Multiple Integrals. Proof of Equations (6.5.6),

(14.3.6)-(14.3.8), and (A.27.6) 446 A19. Determinants of the Forms [Sij — mvj] and [A + Ui — Uj — sgn (i — j)] . . . . 449 A20. Power Series Expansion of Im(9), Equation (6.8.12) 451 A21. Proof of the Inequalities (6.8.15) 452 A22. Proof of Equations (9.1.11) and (9.2.11) 454 A23. Proof of Theorems 10.9.1 and 10.9.2 455 A24. Proof of the Inequality (11.1.5) 464 A25. Good's Proof of Equation (11.1.11) 465 A26. Some Recurrence Relations and Integrals Used in Chapter 14 466 A27. Normalization Integral, Equation (14.1.11) 473 A28. Another Normalization Integral, Equation (14.2.9) 479 A29. Joint Probability Density as a Determinant of a Self-Dual Quaternion Matrix.

Section 14.4, Equations (14.4.2) and (14.4.5) 480 A30. Veriflcation of Equation (14.4.3) 486 A31. The Limits of JN(X, y) and D^(x,i/) as N—>oo. Asymptotic Forms of

J(r;p) and D(r;p). Sections 14.1 and 14.2 488 A32. Evaluation of the Integral (15.1.9) for Complex Matrices 491 A33. A Few Remarks About the Eigenvalues of a Quaternion Real Matrix and Its

Diagonalization 495 A34. Evaluation of the Integral (15.2.9) 498 A35. Another Proof of Equations (15.1.10) and (15.2.10) 501 A36. Proof of Equation (15.2.38) 504 A37. The Case of Random Real Matrices 505 A38. Variance of the Number Statistic. Section 16.1 507 A39. Optimum Linear Statistic. Section 16.1 517 A40. Mean Value of A. Section 16.2 519 A41. Tables of Functions Bß(x\, xi), and Vg(x\, x%) for ß = 1 and 2 525

Notes 535 References 545

Author Index 555

Subject Index 559