free entropy for free gibbs laws given by convex potentialsdavidjekel/ymc_a_free_entropy_talk.pdfwe...

Free Entropy for Free Gibbs Laws Given by ConvexPotentials

David A. Jekel

University of California, Los Angeles

Young Mathematicians in C ∗-algebras,August 2018

David A. Jekel (UCLA) Free Entropy YMC∗A 2018 1 / 24

Overview

We will discuss Voiculescu’s free entropy of a non-commutative law µ ofan m-tuple. This is an analogue in free probability theory of thecontinuous entropy of a probability measure.

Voiculescu defined two types of free entropy, χ(µ) and χ∗(µ). They bothmeasure the “regularity” of the law µ. They are based on two differentviewpoints for classical entropy.

Theorem (Biane, Capitaine, Guionnet 2003)

χ(µ) ≤ χ∗(µ).

Overview

χ(µ) ≤ χ∗(µ).

Overview

χ(µ) ≤ χ∗(µ).

Overview

Theorem (Dabrowski 2017, J. 2018)

Suppose that µ is a free Gibbs state given by a nice enough convexpotential V . Then

χ(µ) = χ∗(µ).

The strategy is to approximate µ by N × N random matrix models, andget the free theory as a limit of the classical theory.

Our new proof is “more elementary” in the sense that it doesn’t use SDE,just basic PDE tools.

Overview

χ(µ) = χ∗(µ).

Overview

χ(µ) = χ∗(µ).

Free Probability and Non-Commutative Laws

What is non-commutative probability?

classical non-commutative

L∞(Ω,P) W ∗-algebra Mexpectation E trace τ

bdd. real rand. var. X self-adjoint X ∈ Mlaw of X spectral distribution of X w.r.t. τ

What is free probability?

We replace classical independence by free independence.

Definition by Example

For groups G1 and G2, the algebras L(G1) and L(G2) are freelyindependent in (L(G1 ∗ G2), τ).

Free Central Limit Theorem: There’s a free central limit theorem withnormal distribution replaced by semicircular distribution.

Free Convolution: If X and Y are classically independent, thenµX+Y = µX ∗ µY . If X and Y are freely independent, thenµX+Y = µX µY .

What is the law of a tuple?

Classically, the law of X = (X1, . . . ,Xm) is a measure on Rm given by

µX (A) = P(X ∈ A).

Assuming finite moments, this can be viewed as a map

µX : C[x1, . . . , xm]→ C, p(x1, . . . , xm) 7→ E [p(X1, . . . ,Xm)].

In the non-commutative case, the law of X = (X1, . . . ,Xm) ∈ Mmsa is

defined as the map

µX : C〈x1, . . . , xm〉 → C, p(x1, . . . , xm) 7→ τ [p(X1, . . . ,Xm)],

The moment topology on laws is given by pointwise convergence onC〈x1, . . . , xm〉.

µX (A) = P(X ∈ A).

defined as the map

µX (A) = P(X ∈ A).

defined as the map

µX (A) = P(X ∈ A).

defined as the map

Microstates Free Entropy χ

What is classical entropy?

The continuous entropy of a probability measure dµ(x) = ρ(x) dx on Rm

is given by

h(µ) = −∫ρ log ρ.

If µ does not have a density, we set h(µ) = −∞.

“Entropy measures regularity.”

1 If µ is highly concentrated, then there is large negative entropy.

2 For mean zero and variance 1, the highest entropy is achieved byGaussian.

3 If you smooth µ out by convolution, the entropy increases.

What is classical entropy?

The continuous entropy of a probability measure dµ(x) = ρ(x) dx on Rm

is given by

h(µ) = −∫ρ log ρ.

If µ does not have a density, we set h(µ) = −∞.

“Entropy measures regularity.”

1 If µ is highly concentrated, then there is large negative entropy.

2 For mean zero and variance 1, the highest entropy is achieved byGaussian.

3 If you smooth µ out by convolution, the entropy increases.

Microstates Interpretation

Since there’s no nice integral formula for entropy in the free case, thedefinition of χ is based on the microstates interpretation.

Classical case: Given a vector in x = (x1, . . . , xm) ∈ (RN)m, let’s define itsempirical distribution as

µx =1

N∑j=1

δ((x1)j ,...,(xm)j ).

Then x : µx is close to µ has measure approximately exp(−Nh(µ)).Thus, h(µ) can be expressed as

inf(nbhd’s of µ)

limN→∞

Nlog volx : µx close to µ.

Intuition: If µ is more regular and spread out, then there are moremicrostates because most choices of N vectors are “evenly distributed.”

µx =1

N∑j=1

δ((x1)j ,...,(xm)j ).

Then x : µx is close to µ has measure approximately exp(−Nh(µ)).

Thus, h(µ) can be expressed as

inf(nbhd’s of µ)

limN→∞

µx =1

N∑j=1

δ((x1)j ,...,(xm)j ).

inf(nbhd’s of µ)

limN→∞

µx =1

N∑j=1

δ((x1)j ,...,(xm)j ).

inf(nbhd’s of µ)

limN→∞

Microstates Free Entropy

Idea for free case: Replace RN (self-adjoints in L∞(1, . . . ,N)) byMN(C)sa.

Given (x1, . . . , xm) ∈ MN(C)m, the empirical distribution µx is thenon-commutative law of x w.r.t. normalized trace on MN(C). Define

χ(µ) = inf(nbhd’s of µ)

lim supN→∞

N2log volx : µx close to µ,

where the “closeness” is defined in the moment topology.

(Voiculescu) χ has properties similar to h, and also relates to properties ofthe W ∗ algebra generated by a tuple with the law µ.

lim supN→∞

Non-microstates Free Entropy χ∗

Classical Fisher Information

Classical case: Let µ be a probability measure on Rm with density ρ. Letγt be the law of a Gaussian random vector with variance tI . Then

dth(µ ∗ γt) =

∫|∇ρt |2/ρt = ‖∇ρt/ρt‖2L2(µ∗γt).

The quantity ‖∇ρt/ρt‖2L2(µ∗γt) is called the Fisher information of µ ∗ γt .The entropy can be recovered by integrating the Fisher information.

Intuition: The Fisher information measures the regularity of µ by lookingat its derivatives.

dth(µ ∗ γt) =

∫|∇ρt |2/ρt = ‖∇ρt/ρt‖2L2(µ∗γt).

dth(µ ∗ γt) =

∫|∇ρt |2/ρt = ‖∇ρt/ρt‖2L2(µ∗γt).

Non-Microstates Free Entropy

In the free case, there’s no density, so we want to rephrase the definitionusing integration by parts.

Classical Fisher information is L2 norm of the conjugate variableξ = (∇ρ/ρ)(X ), which is characterized by an integration-by-parts formulaE [ξf (X )] = E [∇f (X )].

Voiculescu used a free version of this integration-by-parts formula in orderto define the free conjugate variables and hence the free Fisherinformation.

χ∗(µ) is defined by integrating the free Fisher information of µ σt ,where σt is semicircular.

Free Gibbs Laws

Let V be a nice convex function on MN(C)msa, such as

V (x) =1

2‖x‖22 + εRe τ [f ((x1 + i)−1, . . . , (xm + i)−1)],

for x = (x1, . . . , xm), where f is a non-commutative ∗-polynomial, τ isnormalized trace, ‖x‖22 =

∑j τ(x2j ), and ε is small.

Define probability measure on MN(C)msa) by

dµN(x) =1

ZNexp(−N2V (x)) dx ,

where ZN is a normalizing constant.

Free Gibbs Laws

Let V be a nice convex function on MN(C)msa, such as

V (x) =1

2‖x‖22 + εRe τ [f ((x1 + i)−1, . . . , (xm + i)−1)],

for x = (x1, . . . , xm), where f is a non-commutative ∗-polynomial, τ isnormalized trace, ‖x‖22 =

∑j τ(x2j ), and ε is small.

Define probability measure on MN(C)msa) by

dµN(x) =1

ZNexp(−N2V (x)) dx ,

where ZN is a normalizing constant.

Free Gibbs Laws

Theorem (Dabrowski, Guionnet, Shlyakhtenko, . . . )

For non-commutative polynomials p, the limN→∞ EµN [τ(p(X ))] exists.Also, τ [p(X )] is close to the average with high probability under µN(“concentration of measure”).

Definition

The free Gibbs law given by V is the law µ(p) = limN→∞ EµN [τ(p(X ))].

Free Gibbs Laws

Theorem (Dabrowski, Guionnet, Shlyakhtenko, . . . )

For non-commutative polynomials p, the limN→∞ EµN [τ(p(X ))] exists.Also, τ [p(X )] is close to the average with high probability under µN(“concentration of measure”).

Definition

The free Gibbs law given by V is the law µ(p) = limN→∞ EµN [τ(p(X ))].

Some of the Proof

Strategy

We want to show that χ(µ) is the integral of the free Fisher information ofµ σt .

From classical theory, h(µN) is the integral of the Fisher information forµN ∗ σt,N , where σt,N is the law of Gaussian random matrix (GUE).

It suffices to show that h(µN) (normalized) converges to χ(µ) and theclassical Fisher information of µN ∗ σt,N converges to the free Fisherinformation for µ σt .

Strategy

Two “Easy” Lemmas

Lemma 1

χ(µ) is the lim sup of h(µN) (renormalized).

Proof.

Due to concentration of measure, µN is a good approximation of theuniform distribution on a microstate space, so the entropy of µN is a goodapproximation of the log volume of the microstate space.

Lemma 2

The Fisher information of µN converges to the free Fisher information of µ.

Proof.

For the measure µN , the conjugate variable ∇ρ/ρ is simply DV(renormalized). The integration-by-parts formula passes to the limit, soDV is the free conjugate variable, and ‖DV ‖L2(µN) → ‖DV ‖L2(µ).

Two “Easy” Lemmas

Lemma 1

χ(µ) is the lim sup of h(µN) (renormalized).

Proof.

Due to concentration of measure, µN is a good approximation of theuniform distribution on a microstate space, so the entropy of µN is a goodapproximation of the log volume of the microstate space.

Lemma 2

The Fisher information of µN converges to the free Fisher information of µ.

Proof.

For the measure µN , the conjugate variable ∇ρ/ρ is simply DV(renormalized). The integration-by-parts formula passes to the limit, soDV is the free conjugate variable, and ‖DV ‖L2(µN) → ‖DV ‖L2(µ).

Strategy 2

We need Lemma 2 to work for µN ∗ σt,N , not just for µN .

This measure is given by a potential VN,t , which is not given by exactlythe same formula for all N.

So we want to show that VN,t still has “good asymptotic behavior” asN →∞.

Strategy 2

Approximation Property

Definition

A sequence of functions φN : MN(C)msa → MN(C) is asymptoticallyapproximable by non-commutative polynomials if for every ε > 0 andR > 0, there exists a polynomial p such that

lim supN→∞

supx∈MN(C)msa‖xj‖≤R

‖φN(x)− f (x)‖2 ≤ ε.

If VN is a sequence of convex potentials and DVN has the approximationproperty, then Lemma 2 still works.

Evolution of Potentials

DVN = DV has the approximation property, and we want to show DVN,t

has the approximation property.

The potential evolves according to

∂tVN,t =1

2N∆VN,t −

2‖DVN,t‖22

Using PDE tools and the convexity assumptions, we can “build” anapproximate solution from VN using operations that preserve theapproximation property (including convolution with GUE, composition,and limits).

∂tVN,t =1

2N∆VN,t −

2‖DVN,t‖22

∂tVN,t =1

2N∆VN,t −

2‖DVN,t‖22

Philosophical Remarks

The strategy of proving χ = χ∗ by taking the limit of finite-dimensionalclassical models is “obvious.” The question is how to carry it outtechnically . . .

Like getting a camel to walk backwards, it’s not apparent to an observerwhy this is so hard. But there are other situations where the convergenceof finite-dimensional models to the idealized infinite limit is not as nice(e.g. higher-dimensional Coulomb gas — Sylvia Serfaty).

The asymptotic approximation property is a way to make precise the ideaof having a well-defined and free-probabilistic limiting behavior. This couldhave other applications.

Thank you!!

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