The norm of polynomials in large random matrices

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, 11-21 April 2011. Camille Mâle (ENS de Lyon) The norm of polynomials in large RM 1 / 23

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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