Free Probability and Random Matrices: from isomorphisms to universality

Transcription

1 Free Probability and Random Matrices: from isomorphisms to universality Alice Guionnet MIT TexAMP, November 21, 2014 Joint works with F. Bekerman, Y. Dabrowski, A.Figalli, E. Maurel-Segala, J. Novak, D. Shlyakhtenko.

2 Free Probability Classical Probability Operator algebra Commutative algebra Large Random Matrices

3 Free probability Classical Probability Isomorphisms between Transport maps C and W algebras Optimal transport Random Matrices Universality

4 Outline Free probability Random matrices Transport maps The isomorphism problem Proofs : Monge-Ampère equation Approximate transport and universality

5 Free probability and Random matrices Free probability Random matrices Transport maps The isomorphism problem Proofs : Monge-Ampère equation Approximate transport and universality

6 Free probability theory = Non-Commutative probability theory +Notion of Freeness

7 What is a non-commutative law?

8 What is a non-commutative law? What is a classical law on R d?

9 What is a non-commutative law? What is a classical law on R d? It is a linear map Q : f C b (R d, R) Q(f ) = f (x)dq(x) R

10 What is a non-commutative law? What is a classical law on R d? It is a linear map Q : f C b (R d, R) Q(f ) = f (x)dq(x) R A non-commutative law τ of n self-adjoint variables is a linear map τ : P C X 1,, X d τ(p) C

11 What is a non-commutative law? What is a classical law on R d? It is a linear map Q : f C b (R d, R) Q(f ) = f (x)dq(x) R A non-commutative law τ of n self-adjoint variables is a linear map τ : P C X 1,, X d τ(p) C It should satisfy Positivity : τ(pp ) 0 for all P, (zx i1 X ik ) = zx ik X i1, Mass : τ(1) = 1, Traciality : τ(pq) = τ(qp) for all P, Q.

12 Free probability : non-commutative law+freeness X = (X 1,..., X m ) and Y = (Y 1,, Y n ) are free under τ iff for all polynomials P 1... P l and Q 1, Q l so that τ(p i (X)) = 0 and τ(q i (Y)) = 0 τ (P 1 (X)Q 1 (Y) P l (X)Q l (Y)) = 0.

13 Free probability : non-commutative law+freeness X = (X 1,..., X m ) and Y = (Y 1,, Y n ) are free under τ iff for all polynomials P 1... P l and Q 1, Q l so that τ(p i (X)) = 0 and τ(q i (Y)) = 0 τ (P 1 (X)Q 1 (Y) P l (X)Q l (Y)) = 0. τ is uniquely determined by τ(p(x)) and τ(q(y)), Q, P polynomials.

14 Free probability : non-commutative law+freeness X = (X 1,..., X m ) and Y = (Y 1,, Y n ) are free under τ iff for all polynomials P 1... P l and Q 1, Q l so that τ(p i (X)) = 0 and τ(q i (Y)) = 0 τ (P 1 (X)Q 1 (Y) P l (X)Q l (Y)) = 0. τ is uniquely determined by τ(p(x)) and τ(q(y)), Q, P polynomials. Let G be a group with free generators g 1,, g m, neutral e τ(g) = 1 g=e, for g G is the law of m free variables.

15 Laws and representations as bounded linear operators Let τ be a non-commutative law, that is a linear form on C X 1,..., X d so that τ(pp ) 0, τ(1) = 1, τ(pq) = τ(qp), which is bounded, i.e. and for all i k {1,..., d}, all l N, τ(x i1 X il ) R l. By the Gelfand-Naimark-Segal construction, we can associate to τ a Hilbert space H, Ω H, and a 1,..., a d bounded linear operators on H so that for all P τ(p(x 1,..., X d )) = Ω, P(a 1,..., a d )Ω H.

16 Free probability and Random matrices Free probability Random matrices Transport maps The isomorphism problem Proofs : Monge-Ampère equation Approximate transport and universality

17 Examples of non-commutative laws : τ linear, τ(pp ) 0, τ(1) = 1, τ(pq) = τ(qp) Let (X1 N,, X d N ) be d N N Hermitian random matrices, τ X N (P) := E[ 1 ( N Tr P(X1 N,, Xd )] N ) Here Tr(A) = N i=1 A ii.

18 Examples of non-commutative laws : τ linear, τ(pp ) 0, τ(1) = 1, τ(pq) = τ(qp) Let (X1 N,, X d N ) be d N N Hermitian random matrices, τ X N (P) := E[ 1 ( N Tr P(X1 N,, Xd )] N ) Here Tr(A) = N i=1 A ii. Let (X1 N,, X d N ) be d N N Hermitian random matrices for N 0 so that τ(p) := lim E[ 1 ( N N Tr P(X1 N,, Xd )] N ) exists for all polynomial P.

19 The Gaussian Unitary Ensemble X N follows the GUE iff it is a N N matrix so that (X N ) = X N, (X N kl ) k l are independent, With g kl, g kl iid centered Gaussian variables with variance one X N kl = 1 2N (g kl + i g kl ), k < l, X N kk = 1 N g kk In other words, the law of the GUE is given by dp(x N ) = 1 Z N exp{ N 2 Tr((X N ) 2 )}dx N with dx N = k l drx kl N k<l dix kl N.

20 The GUE and the semicircle law Let X N be a matrix following the Gaussian Unitary Ensemble, that is a dp(x N ) = 1 Z N exp{ N 2 Tr((X N ) 2 )}dx N Theorem (Wigner 58 ) With σ the semicircle distribution, lim E[ 1 N N Tr((X N ) p )] = x p σ(dx) p N x p dσ(x) is the number of non-crossing pair partitions of p points.

21 Let X N 1,, X N d The law of free semicircle variables be independent GUE matrices, that is ( ) P dx1 N,, dxd N = 1 d (Z N ) d exp{ N 2 Tr( Theorem (Voiculescu 91 ) For any polynomial P C X 1,, X d i=1 (X N lim E[ 1 N N Tr(P(X 1 N,, Xd N ))] = σ(p) i ) 2 )} dx N i. σ is the law of d free semicircle variables. If P = X i1 X i2 X ik, σ(p) is the number of non-crossing color wise pair partitions build over points of color i 1, i 2...

22 More general laws Let V C X 1,..., X d and set P N V (dx N 1,..., dx N d ) = 1 Z N V exp{ NTr(V (X N 1,..., X N d ))}dx N 1 dx N d Theorem ( G Maurel Segala 06 and G Shlyakhtenko 09 ) Assume that V satisfies a local convexity property [e.g V = 2 1 Xi 2 + W, W small]. Then there exists a non-commutative law τ V so that for any polynomial P 1 τ V (P) = lim N N Tr(P(X 1 N,..., Xd N ))dpn V (X 1 N,..., Xd N )

23 Example ; q-gaussian variables [Bozejko and Speicher 91 ] A d-tuple of q-gaussian variables is such that τ q,d (X i1 X ip ) = π q i(π) i k {1,, d} where the sum runs over pair partitions of colored dots whose block contains dots of the same color and i(π) is the number of crossings. i(π) = 4

24 Example ; q-gaussian variables [Bozejko and Speicher 91 ] A d-tuple of q-gaussian variables is such that τ q,d (X i1 X ip ) = π q i(π) i k {1,, d} where the sum runs over pair partitions of colored dots whose block contains dots of the same color and i(π) is the number of crossings. i(π) = 4 Theorem (Dabrowski 10 ) If dq small, there exists V q,d = 1/2 Xi 2 + W q,d with W q,d small so that 1 τ q,d (P) = lim N N Tr(P(X 1 N,..., Xd N ))dpn V q,d (X1 N,..., Xd N )

25 Free probability and Random matrices Free probability Random matrices Transport maps The isomorphism problem Proofs : Monge-Ampère equation Approximate transport and universality

26 The isomorphism problem Let τ, µ be two non-commutative laws of d (resp. m) variables X = (X 1,..., X d ) (resp. Y = (Y 1,..., Y m )). Can we find transport maps T = (T 1,..., T m ) and T = (T 1,..., T d ) of d (resp. m) variables so that for all polynomials P, Q τ(p(x )) = µ(p(t 1 (Y ),..., T d (Y ))) µ(q(y )) = τ(q(t 1(X ),..., T m(x ))) We denote τ = T µ and µ = T τ

27 The isomorphism problem Let τ, µ be two non-commutative laws of d (resp. m) variables X = (X 1,..., X d ) (resp. Y = (Y 1,..., Y m )). Can we find transport maps T = (T 1,..., T m ) and T = (T 1,..., T d ) of d (resp. m) variables so that for all polynomials P, Q τ(p(x )) = µ(p(t 1 (Y ),..., T d (Y ))) µ(q(y )) = τ(q(t 1(X ),..., T m(x ))) We denote τ = T µ and µ = T τ The free group isomorphism problem : Does there exists transport maps from τ to µ, the law of d (resp. m) free variables with d m?

28 Classical transport Let P, Q be two probability measures on R d and R m respectively. A transport map from P to Q is a measurable function T : R d R m so that for all bounded continuous function f f (T (x))dp(x) = f (x)dq(x). T We denote T #P = Q. P Q Fact (von Neumann [1932]) : If P, Q dx, T exists.

29 Free transport (G. and Shlyakhtenko 12 ) Recall that 1 τ W (P) = lim N N Tr(P(X 1 N,..., Xd N ))dpn V (X 1 N,..., Xd N ) with V = 1 2 X 2 i + W with W self-adjoint, small Theorem There exists F W, T W smooth transport maps between τ W, σ = τ 0 so that for all polynomial P τ W = T W τ 0 τ 0 = F W τ W In particular the related C algebras and von Neumann algebras are isomorphic.

30 Isomorphisms of q-gaussian algebras Let τ q,d be the law of d q Gaussian τ q,d (X i1 X ip ) = π q i(π) i k {1,, d} Theorem (G Shlyakhtenko 12 ) For qd small enough, there exists smooth transport maps between τ q,d and τ q,0 = σ. In particular the C -algebra and von Neumann algebras of q-gaussian laws, q small, are isomorphic to that of free semicircle law σ.

31 Idea of the proof : Monge-Ampère equation Let P, Q be probability measures on R d that have smooth densities P(dx) = e V (x) dx Q(dx) = e W (x) dx. Then T #P = Q is equivalent to f (T (x))e V (x) dx = = f (x)e W (x) dx f (T (y))e W (T (y)) JT (y)dy with JT the Jacobian of T. Hence, it is equivalent to the Monge-Ampère equation V (x) = W (T (x)) log JT (x).

32 Idea of the proof : Monge-Ampère equation Let P, Q be probability measures on R d that have smooth densities P(dx) = e V (x) dx Q(dx) = e W (x) dx. Then T #P = Q is equivalent to f (T (x))e V (x) dx = = f (x)e W (x) dx f (T (y))e W (T (y)) JT (y)dy with JT the Jacobian of T. Hence, it is equivalent to the Monge-Ampère equation V (x) = W (T (x)) log JT (x). There exists a free analogue to Monge-Ampère equation. For W V small it has a unique solution.

33 Non-perturbative transport maps P N V (dx 1 N,..., dxd N ) = 1 e NTr(V (X 1 N,...,X N)) d dxi N Z N 1 τ W (P) = lim N N Tr(P(X 1 N,..., Xd N ))dpn 1 2 X 2 i +W (X 1 N,..., Xd N ) Theorem (WIP with Y-Dabrowski and D-Shlyakhtenko 14 ) Assume that V = 1 2 X 2 i + W is strictly convex, then there exists (F i ) 1 i d (C X 1,..., X d ) d so that τ W = F #τ 0.

34 The Poisson approach Transport µ V (dx) = e V (x) dx to µ W (dx) = e W (x) dx by interpolation. Define a flow T s,t so that T s,t #µ Vs = µ Vt, V t = (1 t)v + tw. Let φ t = lim s t T s,t = t T 0,t T 1 0,t (1) If φ t = ψ t, Monge-Ampère equation becomes L t ψ t = W V (2) with L t = V t. infinitesimal generator. Program : Solve Poisson equation (2) by ψ t = 0 e slt (W V )ds and then deduce T 0,t solution of the transport equation (1) driven by ψ t.

35 The Poisson approach Transport µ V (dx) = e V (x) dx to µ W (dx) = e W (x) dx by interpolation. Define a flow T s,t so that T s,t #µ Vs = µ Vt, V t = (1 t)v + tw. Let φ t = lim s t T s,t = t T 0,t T 1 0,t (1) If φ t = ψ t, Monge-Ampère equation becomes L t ψ t = W V (2) with L t = V t. infinitesimal generator. Program : Solve Poisson equation (2) by ψ t = 0 e slt (W V )ds and then deduce T 0,t solution of the transport equation (1) driven by ψ t. Generalizes by using free Stochastic differential equation.

36 Free probability and Random matrices Free probability Random matrices Transport maps The isomorphism problem Proofs : Monge-Ampère equation Approximate transport and universality

37 dp N β,v (λ 1,..., λ N ) = 1 Z N β,v Local fluctuations in RMT 1 λ1 <λ 2 < <λ N λ i λ j β e Nβ λ 2 i dλi For β = 1, 2, 4, Tracy and Widom (93)showed that for each E [ 2, 2], any compactly supported function f lim E P N β,v N [ f (N(λ i1 E),..., N(λ ik E))] = ρ k β (f ) i 1 < <i k Tao showed (12) that N(λ i λ i 1 ) converges towards the Gaudin distribution. Moreover, Tracy -Widom (93) proved i<j lim E P N β,v N [f (N 2/3 (λ N 2))] = TW β (f ).

38 dp N β,v (λ 1,..., λ N ) = 1 Z N β,v Local fluctuations in RMT 1 λ1 <λ 2 < <λ N λ i λ j β e Nβ λ 2 i dλi For β = 1, 2, 4, Tracy and Widom (93)showed that for each E [ 2, 2], any compactly supported function f lim E P N β,v N [ f (N(λ i1 E),..., N(λ ik E))] = ρ k β (f ) i 1 < <i k Tao showed (12) that N(λ i λ i 1 ) converges towards the Gaudin distribution. Moreover, Tracy -Widom (93) proved i<j lim E P N β,v N [f (N 2/3 (λ N 2))] = TW β (f ). For β 0, this was extended by Ramirez,Rider, Virag at the edge (06) and by Valko,Virag (07) at the edge.

39 dp N β,v (λ 1,..., λ N ) = 1 Z N β,v Then Universality for β-models lim N 1 N λ i λ j β e N V (λ i ) dλ i. i<j f (λi ) = f (x)dµ V (x) Theorem ( Bourgade, Erdös, Yau ( , )) Assume β 1, V C 4 (R), µ V with a connected support vanishing as a square root at the boundary, the local fluctuations of the eigenvalues are as in the case V = βx 2.

40 Approximate transport for β-models dp N β,v (λ 1,..., λ N ) = 1 Z N β,v λ i λ j β e N V (λ i ) dλ i i<j Theorem (Bekerman Figalli G 2013) β 0. Assume V, W C 37 (R), with equilibrium measures µ V, µ W with connected support. Assume V, W are off-critical. Then, there exists T 0 : R R C 19, T 1 : R N R N C 1 so that (T0 N + T 1 log N N )#PN β,v PN β,w TV const. N,

41 Approximate transport for β-models dp N β,v (λ 1,..., λ N ) = 1 Z N β,v λ i λ j β e N V (λ i ) dλ i i<j Theorem (Bekerman Figalli G 2013) β 0. Assume V, W C 37 (R), with equilibrium measures µ V, µ W with connected support. Assume V, W are off-critical. Then, there exists T 0 : R R C 19, T 1 : R N R N C 1 so that (T0 N + T 1 log N N )#PN β,v PN β,w TV const. N, P N β,v ( sup T N,k 1 L 1 (P V N 1 k N ) + sup Hence, universality holds. N,k T k,k 1 1 T N,k N λk λ k C log N ) N c.

42 Universality in several matrix models dp V N (X 1 N,..., Xd N ) = 1 ZV N e N 2 Tr(V (X 1 N,...,X d N)) i 1 X N i R dx i N Theorem Assume V = V = 1 2 X 2 i + t i q i, t i small. The law of the spacing Ncj i(λi j λi j+1 ) of the eigenvalues of X N i converges to the Gaudin distribution, and that of N 2/3 c i (max j λ i j C i) to the Tracy-Widom law.

43 Universality for polynomial in several random matrices [Figalli-G 14 ] Let P be a self-adjoint polynomial in d indeterminates and let X1 N,..., X d N be d independent GUE or GOE matrices. Haagerup and Thorbjornsen proved that the largest eigenvalue of P(X1 N,..., X d N ) converges a.s to its free limit.

44 Universality for polynomial in several random matrices [Figalli-G 14 ] Let P be a self-adjoint polynomial in d indeterminates and let X1 N,..., X d N be d independent GUE or GOE matrices. Haagerup and Thorbjornsen proved that the largest eigenvalue of P(X1 N,..., X d N ) converges a.s to its free limit.for ɛ small enough, the eigenvalues of Y N = X N 1 + ɛp(x N 1,..., X N d ) fluctuates locally as when ɛ = 0, that is the spacings follow Gaudin distribution in the bulk and the Tracy-Widom law at the edge.

45 Proof : Beta-models and Monge-Ampère dp N β,v (λ 1,..., λ N ) = 1 Z N β,v λ i λ j β e N V (λ i ) dλ i i<j T #P N β,v = PN β,w satisfies the Monge-Ampère equation PN β,v -a.s. : β i<j log T i(λ) T j (λ) λ i λ j = N (V (T i (λ)) W (λ i ) log Xi T i (λ)/n) If V W small, it can be solved by implicit function theorem.

46 Proof : Beta-models and Monge-Ampère dp N β,v (λ 1,..., λ N ) = 1 Z N β,v λ i λ j β e N V (λ i ) dλ i i<j T #P N β,v = PN β,w satisfies the Monge-Ampère equation PN β,v -a.s. : β i<j log T i(λ) T j (λ) λ i λ j = N (V (T i (λ)) W (λ i ) log Xi T i (λ)/n) If V W small, it can be solved by implicit function theorem. Goal : Take V t = tv + (1 t)w. Define T 0,t #P N β,v 0 = P N β,v t. Then, φ t = t T 0,t T 1 0,t satisfies M t φ t = W V with M t f = β i<j f i (λ) f j (λ) λ i λ j + Xi f i N V (λ i )f i

47 Ansatz and approximate transport Take V t = tv + (1 t)w. Aim : Build φ t, t T 0,t (T 0,t ) 1 = φ t so that R N t = M t φ t (V W ) goes to zero in L 1 (P V N ).Then T 0,t solution of t T 0,t = φ t (T 0,t ) is an approximate transport. Ansatz : φ i t(λ) = φ 0,t (λ i ) + 1 N φ 1,t(λ i ) + 1 φ 2,t (λ i, λ j ). N j

48 Ansatz and approximate transport Take V t = tv + (1 t)w. Aim : Build φ t, t T 0,t (T 0,t ) 1 = φ t so that R N t = M t φ t (V W ) goes to zero in L 1 (P V N ).Then T 0,t solution of t T 0,t = φ t (T 0,t ) is an approximate transport. Ansatz : φ i t(λ) = φ 0,t (λ i ) + 1 N φ 1,t(λ i ) + 1 φ 2,t (λ i, λ j ). N We find with M N = (δ λi µ Vt ) Rt N = N [Ξφ 0,t + W V ](x)dm N (x) + j with Ξf (x) = V t (x)f (x) β Ξ is invertible. Choose φ 0,t, φ 1,t, φ 2,t so that R N t f (x) f (y) dµ Vt (y), x y small.

49 Approximate transport maps for several matrix-models P N V a (dx1 N,..., dxd N ) = 1 ZV N e antr(v (X 1 N,...,X d N)) i e NtrW i (X i ) dx N i. Then,the law of the eigenvalues P N V a of X 1 N,..., X d N under PN V a is P N V a (dλ i j) = IN av (λi j) = 1 Z N V I av N (λi j) d λ i j λ i k β e N W i (λ i j ) dλ i j i=1 j<k e antr(v (UN 1 D(λ1 )(U N 1 ),...,U N d D(λ1 )(U N d ) ) du N 1 du N d

50 Approximate transport maps for several matrix-models P N V a (dx1 N,..., dxd N ) = 1 ZV N e antr(v (X 1 N,...,X d N)) i e NtrW i (X i ) dx N i. Then,the law of the eigenvalues P N V a of X 1 N,..., X d N under PN V a is P N V a (dλ i j) = IN av (λi j) = 1 Z N V I av N (λi j) d λ i j λ i k β e N W i (λ i j ) dλ i j i=1 j<k e antr(v (UN 1 D(λ1 )(U N 1 ),...,U N d D(λ1 )(U N d ) ) du N 1 du N d = (1 + O( 1 N )) exp{(n2 F a 2 + NF a 1 + F a 0 )( 1 N δλ i j, 1 i d)} by G-Novak 13.

51 Conclusion Ideas from classical analysis extend to operator algebra via free probability/random matrices.

52 Conclusion Ideas from classical analysis extend to operator algebra via free probability/random matrices. These ideas are robust : they generalize to type III factors, planar algebras etc[b. Nelson] The main questions are now around smoothness of the transport maps and topology

53 Conclusion Ideas from classical analysis extend to operator algebra via free probability/random matrices. These ideas are robust : they generalize to type III factors, planar algebras etc[b. Nelson] The main questions are now around smoothness of the transport maps and topology The main issue in several matrix models lies in the topological expansion : being able to carry it out in non-pertubative regimes would solve important questions in free probability (convergence of entropy etc)