Free Entropy for Free Gibbs Laws Given by Convex Potentials

Transcription

1 Free Entropy for Free Gibbs Laws Given by Convex Potentials David A. Jekel University of California, Los Angeles Young Mathematicians in C -algebras, August 2018 David A. Jekel (UCLA) Free Entropy YMC A / 24

2 Overview We will discuss Voiculescu s free entropy of a non-commutative law µ of an m-tuple. This is an analogue in free probability theory of the continuous entropy of a probability measure. David A. Jekel (UCLA) Free Entropy YMC A / 24

3 Overview We will discuss Voiculescu s free entropy of a non-commutative law µ of an m-tuple. This is an analogue in free probability theory of the continuous entropy of a probability measure. Voiculescu defined two types of free entropy, χ(µ) and χ (µ). They both measure the regularity of the law µ. They are based on two different viewpoints for classical entropy. David A. Jekel (UCLA) Free Entropy YMC A / 24

4 Overview We will discuss Voiculescu s free entropy of a non-commutative law µ of an m-tuple. This is an analogue in free probability theory of the continuous entropy of a probability measure. Voiculescu defined two types of free entropy, χ(µ) and χ (µ). They both measure the regularity of the law µ. They are based on two different viewpoints for classical entropy. Theorem (Biane, Capitaine, Guionnet 2003) χ(µ) χ (µ). David A. Jekel (UCLA) Free Entropy YMC A / 24

5 Overview Theorem (Dabrowski 2017, J. 2018) Suppose that µ is a free Gibbs state given by a nice enough convex potential V. Then χ(µ) = χ (µ). David A. Jekel (UCLA) Free Entropy YMC A / 24

6 Overview Theorem (Dabrowski 2017, J. 2018) Suppose that µ is a free Gibbs state given by a nice enough convex potential V. Then χ(µ) = χ (µ). The strategy is to approximate µ by N N random matrix models, and get the free theory as a limit of the classical theory. David A. Jekel (UCLA) Free Entropy YMC A / 24

7 Overview Theorem (Dabrowski 2017, J. 2018) Suppose that µ is a free Gibbs state given by a nice enough convex potential V. Then χ(µ) = χ (µ). The strategy is to approximate µ by N N random matrix models, and get 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. David A. Jekel (UCLA) Free Entropy YMC A / 24

8 Free Probability and Non-Commutative Laws David A. Jekel (UCLA) Free Entropy YMC A / 24

9 What is non-commutative probability? classical L (Ω, P) expectation E bdd. real rand. var. X law of X non-commutative W -algebra M trace τ self-adjoint X M spectral distribution of X w.r.t. τ David A. Jekel (UCLA) Free Entropy YMC A / 24

10 What is free probability? We replace classical independence by free independence. David A. Jekel (UCLA) Free Entropy YMC A / 24

11 What is free probability? We replace classical independence by free independence. Definition by Example For groups G 1 and G 2, the algebras L(G 1 ) and L(G 2 ) are freely independent in (L(G 1 G 2 ), τ). David A. Jekel (UCLA) Free Entropy YMC A / 24

12 What is free probability? We replace classical independence by free independence. Definition by Example For groups G 1 and G 2, the algebras L(G 1 ) and L(G 2 ) are freely independent in (L(G 1 G 2 ), τ). Free Central Limit Theorem: There s a free central limit theorem with normal distribution replaced by semicircular distribution. David A. Jekel (UCLA) Free Entropy YMC A / 24

13 What is free probability? We replace classical independence by free independence. Definition by Example For groups G 1 and G 2, the algebras L(G 1 ) and L(G 2 ) are freely independent in (L(G 1 G 2 ), τ). Free Central Limit Theorem: There s a free central limit theorem with normal distribution replaced by semicircular distribution. Free Convolution: If X and Y are classically independent, then µ X +Y = µ X µ Y. If X and Y are freely independent, then µ X +Y = µ X µ Y. David A. Jekel (UCLA) Free Entropy YMC A / 24

14 What is the law of a tuple? Classically, the law of X = (X 1,..., X m ) is a measure on R m given by µ X (A) = P(X A). David A. Jekel (UCLA) Free Entropy YMC A / 24

15 What is the law of a tuple? Classically, the law of X = (X 1,..., X m ) is a measure on R m given by µ X (A) = P(X A). Assuming finite moments, this can be viewed as a map µ X : C[x 1,..., x m ] C, p(x 1,..., x m ) E[p(X 1,..., X m )]. David A. Jekel (UCLA) Free Entropy YMC A / 24

16 What is the law of a tuple? Classically, the law of X = (X 1,..., X m ) is a measure on R m given by µ X (A) = P(X A). Assuming finite moments, this can be viewed as a map µ X : C[x 1,..., x m ] C, p(x 1,..., x m ) E[p(X 1,..., X m )]. In the non-commutative case, the law of X = (X 1,..., X m ) M m sa is defined as the map µ X : C x 1,..., x m C, p(x 1,..., x m ) τ[p(x 1,..., X m )], David A. Jekel (UCLA) Free Entropy YMC A / 24

17 What is the law of a tuple? Classically, the law of X = (X 1,..., X m ) is a measure on R m given by µ X (A) = P(X A). Assuming finite moments, this can be viewed as a map µ X : C[x 1,..., x m ] C, p(x 1,..., x m ) E[p(X 1,..., X m )]. In the non-commutative case, the law of X = (X 1,..., X m ) M m sa is defined as the map µ X : C x 1,..., x m C, p(x 1,..., x m ) τ[p(x 1,..., X m )], The moment topology on laws is given by pointwise convergence on C x 1,..., x m. David A. Jekel (UCLA) Free Entropy YMC A / 24

18 Microstates Free Entropy χ David A. Jekel (UCLA) Free Entropy YMC A / 24

19 What is classical entropy? The continuous entropy of a probability measure dµ(x) = ρ(x) dx on R m is given by h(µ) = ρ log ρ. If µ does not have a density, we set h(µ) =. David A. Jekel (UCLA) Free Entropy YMC A / 24

20 What is classical entropy? The continuous entropy of a probability measure dµ(x) = ρ(x) dx on R m 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 by Gaussian. 3 If you smooth µ out by convolution, the entropy increases. David A. Jekel (UCLA) Free Entropy YMC A / 24

21 Microstates Interpretation Since there s no nice integral formula for entropy in the free case, the definition of χ is based on the microstates interpretation. David A. Jekel (UCLA) Free Entropy YMC A / 24

22 Microstates Interpretation Since there s no nice integral formula for entropy in the free case, the definition of χ is based on the microstates interpretation. Classical case: Given a vector in x = (x 1,..., x m ) (R N ) m, let s define its empirical distribution as µ x = 1 N N δ ((x1 ) j,...,(x m) j ). j=1 Then {x : µ x is close to µ} has measure approximately exp( Nh(µ)). David A. Jekel (UCLA) Free Entropy YMC A / 24

23 Microstates Interpretation Since there s no nice integral formula for entropy in the free case, the definition of χ is based on the microstates interpretation. Classical case: Given a vector in x = (x 1,..., x m ) (R N ) m, let s define its empirical distribution as µ x = 1 N N δ ((x1 ) j,...,(x m) j ). j=1 Then {x : µ x is close to µ} has measure approximately exp( Nh(µ)). Thus, h(µ) can be expressed as inf lim (nbhd s of µ) N 1 N log vol{x : µ x close to µ}. David A. Jekel (UCLA) Free Entropy YMC A / 24

24 Microstates Interpretation Since there s no nice integral formula for entropy in the free case, the definition of χ is based on the microstates interpretation. Classical case: Given a vector in x = (x 1,..., x m ) (R N ) m, let s define its empirical distribution as µ x = 1 N N δ ((x1 ) j,...,(x m) j ). j=1 Then {x : µ x is close to µ} has measure approximately exp( Nh(µ)). Thus, h(µ) can be expressed as inf lim (nbhd s of µ) N 1 N log vol{x : µ x close to µ}. Intuition: If µ is more regular and spread out, then there are more microstates because most choices of N vectors are evenly distributed. David A. Jekel (UCLA) Free Entropy YMC A / 24

25 Microstates Free Entropy Idea for free case: Replace R N (self-adjoints in L ({1,..., N})) by M N (C) sa. David A. Jekel (UCLA) Free Entropy YMC A / 24

26 Microstates Free Entropy Idea for free case: Replace R N (self-adjoints in L ({1,..., N})) by M N (C) sa. Given (x 1,..., x m ) M N (C) m, the empirical distribution µ x is the non-commutative law of x w.r.t. normalized trace on M N (C). Define χ(µ) = 1 inf lim sup (nbhd s of µ) N N 2 log vol{x : µ x close to µ}, where the closeness is defined in the moment topology. David A. Jekel (UCLA) Free Entropy YMC A / 24

27 Microstates Free Entropy Idea for free case: Replace R N (self-adjoints in L ({1,..., N})) by M N (C) sa. Given (x 1,..., x m ) M N (C) m, the empirical distribution µ x is the non-commutative law of x w.r.t. normalized trace on M N (C). Define χ(µ) = 1 inf lim sup (nbhd s of µ) N N 2 log vol{x : µ x close to µ}, where the closeness is defined in the moment topology. (Voiculescu) χ has properties similar to h, and also relates to properties of the W algebra generated by a tuple with the law µ. David A. Jekel (UCLA) Free Entropy YMC A / 24

28 Non-microstates Free Entropy χ David A. Jekel (UCLA) Free Entropy YMC A / 24

29 Classical Fisher Information Classical case: Let µ be a probability measure on R m with density ρ. Let γ t be the law of a Gaussian random vector with variance ti. Then d dt h(µ γ t) = ρ t 2 /ρ t = ρ t /ρ t 2 L 2 (µ γ. t) David A. Jekel (UCLA) Free Entropy YMC A / 24

30 Classical Fisher Information Classical case: Let µ be a probability measure on R m with density ρ. Let γ t be the law of a Gaussian random vector with variance ti. Then d dt h(µ γ t) = ρ t 2 /ρ t = ρ t /ρ t 2 L 2 (µ γ. t) The quantity ρ t /ρ t 2 L 2 (µ γ t) is called the Fisher information of µ γ t. The entropy can be recovered by integrating the Fisher information. David A. Jekel (UCLA) Free Entropy YMC A / 24

31 Classical Fisher Information Classical case: Let µ be a probability measure on R m with density ρ. Let γ t be the law of a Gaussian random vector with variance ti. Then d dt h(µ γ t) = ρ t 2 /ρ t = ρ t /ρ t 2 L 2 (µ γ. t) The quantity ρ t /ρ t 2 L 2 (µ γ t) is called the Fisher information of µ γ t. The entropy can be recovered by integrating the Fisher information. Intuition: The Fisher information measures the regularity of µ by looking at its derivatives. David A. Jekel (UCLA) Free Entropy YMC A / 24

32 Non-Microstates Free Entropy In the free case, there s no density, so we want to rephrase the definition using integration by parts. David A. Jekel (UCLA) Free Entropy YMC A / 24

33 Non-Microstates Free Entropy In the free case, there s no density, so we want to rephrase the definition using integration by parts. Classical Fisher information is L 2 norm of the conjugate variable ξ = ( ρ/ρ)(x ), which is characterized by an integration-by-parts formula E[ξf (X )] = E[ f (X )]. David A. Jekel (UCLA) Free Entropy YMC A / 24

34 Non-Microstates Free Entropy In the free case, there s no density, so we want to rephrase the definition using integration by parts. Classical Fisher information is L 2 norm of the conjugate variable ξ = ( ρ/ρ)(x ), which is characterized by an integration-by-parts formula E[ξf (X )] = E[ f (X )]. Voiculescu used a free version of this integration-by-parts formula in order to define the free conjugate variables and hence the free Fisher information. David A. Jekel (UCLA) Free Entropy YMC A / 24

35 Non-Microstates Free Entropy In the free case, there s no density, so we want to rephrase the definition using integration by parts. Classical Fisher information is L 2 norm of the conjugate variable ξ = ( ρ/ρ)(x ), which is characterized by an integration-by-parts formula E[ξf (X )] = E[ f (X )]. Voiculescu used a free version of this integration-by-parts formula in order to define the free conjugate variables and hence the free Fisher information. χ (µ) is defined by integrating the free Fisher information of µ σ t, where σ t is semicircular. David A. Jekel (UCLA) Free Entropy YMC A / 24

36 Free Gibbs Laws David A. Jekel (UCLA) Free Entropy YMC A / 24

37 Free Gibbs Laws Let V be a nice convex function on M N (C) m sa, such as V (x) = 1 2 x ɛ Re τ[f ((x 1 + i) 1,..., (x m + i) 1 )], for x = (x 1,..., x m ), where f is a non-commutative -polynomial, τ is normalized trace, x 2 2 = j τ(x j 2 ), and ɛ is small. David A. Jekel (UCLA) Free Entropy YMC A / 24

38 Free Gibbs Laws Let V be a nice convex function on M N (C) m sa, such as V (x) = 1 2 x ɛ Re τ[f ((x 1 + i) 1,..., (x m + i) 1 )], for x = (x 1,..., x m ), where f is a non-commutative -polynomial, τ is normalized trace, x 2 2 = j τ(x j 2 ), and ɛ is small. Define probability measure on M N (C) m sa) by where Z N is a normalizing constant. dµ N (x) = 1 Z N exp( N 2 V (x)) dx, David A. Jekel (UCLA) Free Entropy YMC A / 24

39 Free Gibbs Laws Theorem (Dabrowski, Guionnet, Shlyakhtenko,... ) For non-commutative polynomials p, the lim N E µn [τ(p(x ))] exists. Also, τ[p(x )] is close to the average with high probability under µ N ( concentration of measure ). David A. Jekel (UCLA) Free Entropy YMC A / 24

40 Free Gibbs Laws Theorem (Dabrowski, Guionnet, Shlyakhtenko,... ) For non-commutative polynomials p, the lim N E µn [τ(p(x ))] exists. Also, τ[p(x )] is close to the average with high probability under µ N ( concentration of measure ). Definition The free Gibbs law given by V is the law µ(p) = lim N E µn [τ(p(x ))]. David A. Jekel (UCLA) Free Entropy YMC A / 24

41 Some of the Proof David A. Jekel (UCLA) Free Entropy YMC A / 24

42 Strategy We want to show that χ(µ) is the integral of the free Fisher information of µ σ t. David A. Jekel (UCLA) Free Entropy YMC A / 24

43 Strategy We want to show that χ(µ) is the integral of the free Fisher information of µ σ t. From classical theory, h(µ N ) is the integral of the Fisher information for µ N σ t,n, where σ t,n is the law of Gaussian random matrix (GUE). David A. Jekel (UCLA) Free Entropy YMC A / 24

44 Strategy We want to show that χ(µ) is the integral of the free Fisher information of µ σ t. From classical theory, h(µ N ) is the integral of the Fisher information for µ N σ t,n, where σ t,n is the law of Gaussian random matrix (GUE). It suffices to show that h(µ N ) (normalized) converges to χ(µ) and the classical Fisher information of µ N σ t,n converges to the free Fisher information for µ σ t. David A. Jekel (UCLA) Free Entropy YMC A / 24

45 Two Easy Lemmas Lemma 1 χ(µ) is the lim sup of h(µ N ) (renormalized). Proof. Due to concentration of measure, µ N is a good approximation of the uniform distribution on a microstate space, so the entropy of µ N is a good approximation of the log volume of the microstate space. David A. Jekel (UCLA) Free Entropy YMC A / 24

46 Two Easy Lemmas Lemma 1 χ(µ) is the lim sup of h(µ N ) (renormalized). Proof. Due to concentration of measure, µ N is a good approximation of the uniform distribution on a microstate space, so the entropy of µ N is a good approximation of the log volume of the microstate space. Lemma 2 The Fisher information of µ N converges to the free Fisher information of µ. Proof. For the measure µ N, the conjugate variable ρ/ρ is simply DV (renormalized). The integration-by-parts formula passes to the limit, so DV is the free conjugate variable, and DV L 2 (µ N ) DV L 2 (µ). David A. Jekel (UCLA) Free Entropy YMC A / 24

47 Strategy 2 We need Lemma 2 to work for µ N σ t,n, not just for µ N. David A. Jekel (UCLA) Free Entropy YMC A / 24

48 Strategy 2 We need Lemma 2 to work for µ N σ t,n, not just for µ N. This measure is given by a potential V N,t, which is not given by exactly the same formula for all N. David A. Jekel (UCLA) Free Entropy YMC A / 24

49 Strategy 2 We need Lemma 2 to work for µ N σ t,n, not just for µ N. This measure is given by a potential V N,t, which is not given by exactly the same formula for all N. So we want to show that V N,t still has good asymptotic behavior as N. David A. Jekel (UCLA) Free Entropy YMC A / 24

50 Approximation Property Definition A sequence of functions φ N : M N (C) m sa M N (C) is asymptotically approximable by non-commutative polynomials if for every ɛ > 0 and R > 0, there exists a polynomial p such that lim sup N sup x M N (C) m sa x j R φ N (x) f (x) 2 ɛ. Lemma If V N is a sequence of convex potentials and DV N has the approximation property, then Lemma 2 still works. David A. Jekel (UCLA) Free Entropy YMC A / 24

51 Evolution of Potentials DV N = DV has the approximation property, and we want to show DV N,t has the approximation property. David A. Jekel (UCLA) Free Entropy YMC A / 24

52 Evolution of Potentials DV N = DV has the approximation property, and we want to show DV N,t has the approximation property. The potential evolves according to t V N,t = 1 2N V N,t 1 2 DV N,t 2 2 David A. Jekel (UCLA) Free Entropy YMC A / 24

53 Evolution of Potentials DV N = DV has the approximation property, and we want to show DV N,t has the approximation property. The potential evolves according to t V N,t = 1 2N V N,t 1 2 DV N,t 2 2 Using PDE tools and the convexity assumptions, we can build an approximate solution from V N using operations that preserve the approximation property (including convolution with GUE, composition, and limits). David A. Jekel (UCLA) Free Entropy YMC A / 24

54 Philosophical Remarks The strategy of proving χ = χ by taking the limit of finite-dimensional classical models is obvious. The question is how to carry it out technically... David A. Jekel (UCLA) Free Entropy YMC A / 24

55 Philosophical Remarks The strategy of proving χ = χ by taking the limit of finite-dimensional classical models is obvious. The question is how to carry it out technically... Like getting a camel to walk backwards, it s not apparent to an observer why this is so hard. But there are other situations where the convergence of finite-dimensional models to the idealized infinite limit is not as nice (e.g. higher-dimensional Coulomb gas Sylvia Serfaty). David A. Jekel (UCLA) Free Entropy YMC A / 24

56 Philosophical Remarks The strategy of proving χ = χ by taking the limit of finite-dimensional classical models is obvious. The question is how to carry it out technically... Like getting a camel to walk backwards, it s not apparent to an observer why this is so hard. But there are other situations where the convergence of 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 idea of having a well-defined and free-probabilistic limiting behavior. This could have other applications. David A. Jekel (UCLA) Free Entropy YMC A / 24

57 Philosophical Remarks The strategy of proving χ = χ by taking the limit of finite-dimensional classical models is obvious. The question is how to carry it out technically... Like getting a camel to walk backwards, it s not apparent to an observer why this is so hard. But there are other situations where the convergence of 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 idea of having a well-defined and free-probabilistic limiting behavior. This could have other applications. Thank you!! David A. Jekel (UCLA) Free Entropy YMC A / 24