The norm of polynomials in large random matrices

Transcription

1 The norm of polynomials in large random matrices Camille Mâle, École Normale Supérieure de Lyon, Ph.D. Student under the direction of Alice Guionnet. with a significant contribution by Dimitri Shlyakhtenko. ESI program Bialgebras in Free Probability, April Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 1 / 23

2 A theorem of strong asymptotic freeness Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 2 / 23

3 The Gaussian Unitary Ensemble (GUE) Definition An N N random matrix X (N) is a GUE matrix if X (N) = X (N) with entries X (N) = (X n,m ) 1 n,m N, where ((X n,n ) 1 n N, ( 2Re (X n,m ), 2Im (X n,m ) ) 1 n<m N ) is a centered Gaussian vector with covariance matrix 1 N 1 N 2. Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 3 / 23

4 The Gaussian Unitary Ensemble (GUE) Definition An N N random matrix X (N) is a GUE matrix if X (N) = X (N) with entries X (N) = (X n,m ) 1 n,m N, where ((X n,n ) 1 n N, ( 2Re (X n,m ), 2Im (X n,m ) ) 1 n<m N ) is a centered Gaussian vector with covariance matrix 1 N 1 N 2. Density 1 ( exp 1 Z N 2N Tr X 2) drex i,j dimx i,j. standard Gaussian measure on i j i<j ( M N (C) Herm, A, B = 1 N Tr [ AB ]). Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 3 / 23

5 The semicircle distribution Definition A non commutative random variable x in a -probability space (A,., τ) has a semicircle distribution of x = x and for any polynomial P, τ [ P(x) ] = Pdσ with dσ(x) = 1 4 x 2π 2 1 x <2 dx. Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 4 / 23

6 The semicircle distribution Definition A non commutative random variable x in a -probability space (A,., τ) has a semicircle distribution of x = x and for any polynomial P, τ [ P(x) ] = Pdσ with dσ(x) = 1 4 x 2π 2 1 x <2 dx. Theorem: Wigner (58), Arnold (67) if X N GUE, then almost surely, for any polynomial P, 1 N Tr[ P(X N ) ] = 1 N N i=1 ( ) P λ i (X N ) N Pdσ= τ [ P(x) ]. Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 4 / 23

7 Voiculescu s asymptotic freeness theorem (1/2) Consider in the -probability space ( M N (C),., τ N := 1 N Tr) X N = (X (N) 1,..., X p (N) ) independent N N GUE matrices, Y N = (Y (N) 1,..., Y (N) q ) N N matrices, independent of X N. Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 5 / 23

8 Voiculescu s asymptotic freeness theorem (1/2) Consider in the -probability space ( M N (C),., τ N := 1 N Tr) X N = (X (N) 1,..., X p (N) ) independent N N GUE matrices, Y N = (Y (N) 1,..., Y (N) q ) N N matrices, independent of X N. Consider in a -probability space (A,., τ) x = (x 1,..., x p ) free semicircular random variables, y = (y 1,..., y q ) free from x. Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 5 / 23

9 Voiculescu s asymptotic freeness theorem (2/2) Theorem: Voiculescu (91), Thorbjørnsen (99), Hiai and Petz (99) Assume Almost surely, Y N L n.c. y when N i.e. for every non commutative polynomial P, τ N [P(Y N, YN )] τ[p(y, N y )], Almost surely, for any j = 1,..., q, one has lim sup N Y (N) j <. Then almost surely (X N, Y N ) Ln.c. (x, y) i.e. for every non commutative polynomial P, τ N [P(X N, Y N, YN )] τ[p(x, y, N y )]. Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 6 / 23

10 C -Probability spaces Definition A C -probability space (A,., τ, ) consists of a -probability space (A,., τ) and a norm such that (A,., ) is a C -algebra. Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 7 / 23

11 C -Probability spaces Definition A C -probability space (A,., τ, ) consists of a -probability space (A,., τ) and a norm such that (A,., ) is a C -algebra. In ( M N (C),., τ N := 1 N Tr) we consider the operator norm: A = ρ(a A) A=A = ρ(a). Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 7 / 23

12 C -Probability spaces Definition A C -probability space (A,., τ, ) consists of a -probability space (A,., τ) and a norm such that (A,., ) is a C -algebra. In ( M N (C),., τ N := 1 N Tr) we consider the operator norm: A = ρ(a A) A=A = ρ(a). Proposition If τ is faithful, i.e. τ[a a] = 0 a = 0, then ( a = lim τ [ (a a) k ] ) 1 2k. k Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 7 / 23

13 A strong asymptotic freeness theorem (1/2) Consider in the C -probability space ( M N (C),., τ N, ) X N = (X (N) 1,..., X p (N) ) independent N N GUE matrices, Y N = (Y (N) 1,..., Y (N) q ) N N matrices, independent of X N. Consider in a C -probability space (A,., τ, ) with faithful trace x = (x 1,..., x p ) free semicircular system, y = (y 1,..., y q ) free from x. Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 8 / 23

14 A strong asymptotic freeness theorem (2/2) Theorem: M. (11), preprint Assume: Almost surely, for every non commutative polynomial P, [ τ N P(YN, YN )] P(Y N, YN ) τ[p(y, y )], N N P(y, y ). Then, almost surely, for every non commutative polynomial P, [ τ N P(XN, Y N, YN )] P(XN, Y N, YN ) τ[p(x, y, y )], N N P(x, y, y ). Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 9 / 23

15 Other results on strong asymptotic freeness Cases where Y N are zeros. Haagerup and Thorbjørnsen (05): pioneering works, Schultz (05): X N GOE, GSE, Capitaine and Donati-Martin (07): X N Wigner ensemble with symmetric law of entries and a concentration assumption; X N Wishart, Anderson (24 Mar 2011 on arxiv): X N Wigner ensemble with finite fourth moment. Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 10 / 23

16 The spectrum of large hermitian matrices Corollary Let H N = P(X N, Y N, YN ) a Hermitian matrix. Denote its empirical eigenvalue distribution L HN = 1 N N δ λi (H N ). i=1 Asymptotic freeness: Almost surely, L HN L h the distribution of the N self adjoint non commutative random variable h = P(x, y, y ). Strong asymptotic freeness: Almost surely, for every ε > 0, there exists N 0 1 such that for every N N 0, Sp ( H N ) Supp ( µ ) + ( ε, ε). Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 11 / 23

17 Proof Idea of the proof Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 12 / 23

18 Proof The main steps Haagerup and Thorbjørnsen s method: 1 A linearization trick, 2 Uniform control of matrix-valued Stieltjes transforms, 3 Concentration argument. Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 13 / 23

19 Proof The main steps Haagerup and Thorbjørnsen s method: 1 A linearization trick, 2 Uniform control of matrix-valued Stieltjes transforms, 3 Concentration argument. In this proof 1 A linearization trick, unchanged, 2 Uniform control of matrix-valued Stieltjes transforms, based on an asymptotic subordination property, 3 An intermediate inclusion of spectrum, by Shlyakhtenko, 4 Concentration argument, no significant changes. Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 13 / 23

20 An equivalent formulation Proof A linearization trick In order to show: Almost surely, for every polynomial P one has P(X N, Y N, YN ) P(x, y, N y ), it is enough to show: Almost surely, for every self adjoint degree one polynomial L with coefficient in M k (C), for any ε > 0, there exists N 0 1 such that for all N N 0, one has Sp ( L(X N, Y N, Y N ) ) Sp ( L(x, y, y ) ) + ( ε, ε). Based on operator spaces techniques (Arveson s theorem and dilation of operators). Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 14 / 23

21 Proof Matricial Stieltjes transforms and R-transforms Let (A,., τ, ) be a C -probability space. Consider z in M k (C) A. Definitions The M k (C)-valued Stieltjes transform of z is G z : M k (C) + M k (C) (Λ ] Λ (id k τ N )[ 1 z) 1. The amalgamated R-transform over M k (C) of z is R z : U M k (C) Λ G z ( 1) (Λ) Λ 1. Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 15 / 23

22 The subordination property Proof Let x = (x 1,..., x p ) and y = (y 1,..., y q ) be selfadjoint elements of A and let a = (a 1,..., a p ) and b = (b 1,..., b q ) be k k Hermitian matrices. Define p q s = a j x j, t = b j y j. Proposition j=1 If the families x and y are free, then one has j=1 G s+t (Λ) = G t (Λ R s ( Gs+t (Λ) ) ). If x 1,..., x p are free semicircular n.c.r.v. then we get p R s : Λ a j Λa j. j=1 Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 16 / 23

23 Proof Stability under analytic perturbations If one has ( G s+t (Λ) = G t (Λ R s Gs+t (Λ) ) ), ( ) G(Λ) = G t (Λ ) R s G(Λ) + Θ(Λ), where Θ is an analytic perturbation, then we get G(Λ) G s+t (Λ) ( 1 + c (Im Λ) 1 2 ) Θ(Λ). Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 17 / 23

24 Proof An asymptotic subordination property Let X N = (X (N) 1,..., X p (N) ) be independent GUE matrices, let Y N = (Y (N) 1,..., Y q (N) ) be deterministic Hermitian matrices and let a = (a 1,..., a p ) and b = (b 1,..., b q ) be k k Hermitian matrices. Define S N = p j=1 a j X (N) j, T N = q j=1 b j Y (N) j. Proposition One has ( G SN +T N (Λ) = G TN (Λ R s GSN +T N (Λ) ) ) + Θ N (Λ), with Θ N an analytic perturbation. Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 18 / 23

25 A first try Proof Let x = (x 1,..., x p ) be free semicircular n.c.r.v. and y = (y 1,..., y q ) the limit in law of Y N. G s+t (Λ) = G t (Λ R s ( Gs+t (Λ) ) ), G SN +T N (Λ) = G TN (Λ R s ( GSN +T N (Λ) ) ) + Θ N (Λ). we get an estimate of G SN +T N (Λ) G s+t (Λ) only if we can control G TN (Λ) G t (Λ). with the concentration machinery we get the Theorem, but with unsatisfactory assumptions on Y N... Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 19 / 23

26 An intermediate space Proof Put x and Y N in a same C -probability space, free from each other. Then G s+tn (Λ) = G TN (Λ R s ( Gs+TN (Λ) ) ), G SN +T N (Λ) = G TN (Λ R s ( GSN +T N (Λ) ) ) + Θ N (Λ). we get an estimate of G SN +T N (Λ) G s+tn (Λ) without any additionnal assumption on Y N. Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 20 / 23

27 Proof An theorem about norm convergence Theorem: by Shlyakhtenko, in an appendix of M. (11) Let Y N = (Y (N) 1,..., Y q (N) ) and y = (y 1,..., y q ) be n.c.r.v. in a C -probability space with faithful trace. Let x = (x 1,..., x q ) be free semicircular n.c.r.v. free from Y N and Y. Assume: for every non commutative polynomial P, [ τ N P(YN, YN )] P(Y N, YN ) τ[p(y, y )], N N P(y, y ), Then for every non commutative polynomial P, P(x, Y N, YN ) P(x, y, N y ). Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 21 / 23

28 Proof An intermediate inclusion of spectrum We get: for every self adjoint degree one polynomial L with coefficient in M k (C), for any ε > 0, there exists N 0 1 such that for all N N 0, one has Sp ( L(x, Y N, Y N ) ) Sp ( L(x, y, y ) ) + ( ε, ε). Together with this estimate of G SN +T N (Λ) G s+tn (Λ), the concentration machinery applies. Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 22 / 23

29 Proof Thank you for your attention Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 23 / 23