Norm of polynomials in Large Random Matrices Camille M ale Ecole - PowerPoint PPT Presentation

Norm of polynomials in Large Random Matrices Camille Mˆ ale ´ Ecole Normale Sup´ erieure de Lyon T´ el´ ecom-Paris Tech, 12 October 2010 Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 1 / 25

Introduction Introduction Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 2 / 25

Introduction The Gaussian Unitary Ensemble (GUE) Definition We said that X ( N ) is an N × N 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 , ( 2 Re ( X n , m ) , 2 Im ( X n , m ) ) 1 � n < m � N is a centered Gaussian vector with covariance matrix 1 N 1 N 2 . Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 3 / 25

Introduction Classical results Let X N ∼ GUE. Denote the eigenvalues of X ( N ) by λ 1 � . . . � λ N . Theorem (Wigner 55) The empirical spectral measure of X ( N ) N L ( X ( N ) ) = 1 � δ λ i N i =1 converges when N → ∞ to the semicircular law with radius 2. Theorem When N → ∞ λ 1 → − 2 , λ N → 2 . Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 4 / 25

Introduction Reformulation Convergence of L ( X ( N ) ) : a.s. and in E in moments N L N ( P ) = 1 P ( λ i ) = 1 � � P ( X ( N ) ) � � N →∞ τ [ P ] := − → Pd σ, N Tr N i =1 √ 4 − t 2 1 | t | � 2 dt the semicircle 1 for all polynomial P , with d σ ( t ) = 2 π distribution. Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 5 / 25

Introduction Reformulation Convergence of L ( X ( N ) ) : a.s. and in E in moments N L N ( P ) = 1 P ( λ i ) = 1 � � P ( X ( N ) ) � � N →∞ τ [ P ] := − → Pd σ, N Tr N i =1 √ 4 − t 2 1 | t | � 2 dt the semicircle 1 for all polynomial P , with d σ ( t ) = 2 π distribution. Convergence of extremal eigenvalues : a.s. � X ( N ) � − N →∞ 2 , → with � · � the operator norm: � ρ ( M ∗ M ) � M � = = ρ ( M ) if M Hermitian where ρ is the spectral radius. Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 5 / 25

Introduction The context of this talk The protagonists X N = ( X ( N ) , . . . , X ( N ) ) family of independent N × N GUE matrices, p 1 Y N = ( Y ( N ) , . . . , Y ( N ) ) family of arbitrary N × N matrices. q 1 Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 6 / 25

Introduction The context of this talk The protagonists X N = ( X ( N ) , . . . , X ( N ) ) family of independent N × N GUE matrices, p 1 Y N = ( Y ( N ) , . . . , Y ( N ) ) family of arbitrary N × N matrices. q 1 We want to extend such results for matrices of the form M N = P ( X N , Y N , Y ∗ N ) , where P is any non commutative polynomial in p + 2 q indeterminates, Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 6 / 25

Introduction The context of this talk The protagonists X N = ( X ( N ) , . . . , X ( N ) ) family of independent N × N GUE matrices, p 1 Y N = ( Y ( N ) , . . . , Y ( N ) ) family of arbitrary N × N matrices. q 1 We want to extend such results for matrices of the form M N = P ( X N , Y N , Y ∗ N ) , where P is any non commutative polynomial in p + 2 q indeterminates, express the asymptotic statistics in elegant terms with m = P ( x , y , y ∗ ) . Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 6 / 25

Free Probability Free Probability Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 7 / 25

Free Probability Non commutative probability space Definition of a ∗ -probability space ( A , · ∗ , τ ) A : unital C -algebra, · ∗ : antilinear involution such that ( ab ) ∗ = b ∗ a ∗ ∀ a , b ∈ A , τ : linear form such that τ [ 1 ] = 1. Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 8 / 25

Free Probability Non commutative probability space Definition of a ∗ -probability space ( A , · ∗ , τ ) A : unital C -algebra, · ∗ : antilinear involution such that ( ab ) ∗ = b ∗ a ∗ ∀ a , b ∈ A , τ : linear form such that τ [ 1 ] = 1. Examples Commutative space: Given a probability space (Ω , F , P ), consider ( L ∞ (Ω , µ ) , ¯ · , E ) Matrix spaces: (M N ( C ) , · ∗ , 1 N Tr ) Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 8 / 25

Free Probability Non commutative probability space Definition of a ∗ -probability space ( A , · ∗ , τ ) A : unital C -algebra, · ∗ : antilinear involution such that ( ab ) ∗ = b ∗ a ∗ ∀ a , b ∈ A , τ : linear form such that τ [ 1 ] = 1. We also assume τ is tracial: τ [ ab ] = τ [ ba ] ∀ a , b ∈ A , τ is a faithful state: τ [ a ∗ a ] ≥ 0 , ∀ a ∈ A and vanishes iff a = 0. A is a C ∗ -algebra: it is equipped with a norm � · � such that � a ∗ a � = � a � 2 = � a ∗ � 2 . Examples Commutative space: Given a probability space (Ω , F , P ), consider ( L ∞ (Ω , µ ) , ¯ · , E ) Matrix spaces: (M N ( C ) , · ∗ , 1 N Tr ) Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 8 / 25

Free Probability Non commutative probability space Definition of a ∗ -probability space ( A , · ∗ , τ ) A : unital C -algebra, · ∗ : antilinear involution such that ( ab ) ∗ = b ∗ a ∗ ∀ a , b ∈ A , τ : linear form such that τ [ 1 ] = 1. We also assume τ is tracial: τ [ ab ] = τ [ ba ] ∀ a , b ∈ A , τ is a faithful state: τ [ a ∗ a ] ≥ 0 , ∀ a ∈ A and vanishes iff a = 0. A is a C ∗ -algebra: it is equipped with a norm � · � such that � a ∗ a � = � a � 2 = � a ∗ � 2 . Examples Commutative space: Given a probability space (Ω , F , P ), consider ( L ∞ (Ω , µ ) , ¯ · , E ) and the infinity norm � · � ∞ , Matrix spaces: (M N ( C ) , · ∗ , 1 N Tr ) with the operator norm � ρ ( M ∗ M ). � M � = Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 8 / 25

Free Probability Non commutative random variables Proposition If a = a ∗ then there exists a compactly supported probability measure µ on � � � R such that ∀ P polynomial τ P ( a ) = Pd µ and � � � � � a � = inf A ≥ 0 � µ [ − A , A ] ) = 1 . � Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 9 / 25

Free Probability Non commutative random variables Proposition If a = a ∗ then there exists a compactly supported probability measure µ on � � � R such that ∀ P polynomial τ P ( a ) = Pd µ and � � � � � a � = inf A ≥ 0 � µ [ − A , A ] ) = 1 . � Definition Elements of A : non commutative random variables (n.c.r.v.), � P ( a , a ∗ ) � Set of numbers τ , ∀ P non commutative polynomial : law of a family a = ( a 1 , . . . , a p ) ∈ A p (generalized moments). L n . c . τ [ P ( a N , a ∗ N →∞ τ [ P ( a , a ∗ )] ∀ P : convergence in law a N N )] − → − → a . Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 9 / 25

Free Probability The relation of freeness Definition of freeness The families of n.c.r.v. a 1 , . . . , a p are free iff ∀ K ∈ N , ∀ P 1 , . . . , P K non commutative polynomials � � P 1 ( a i 1 , a ∗ i 1 ) . . . P K ( a i K , a ∗ τ i K ) = 0 � P k ( a i k , a ∗ � as soon as i 1 � = i 2 � = . . . � = i K and τ i k ) = 0 for k = 1 , . . . , K . Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 10 / 25

Free Probability The relation of freeness Definition of freeness The families of n.c.r.v. a 1 , . . . , a p are free iff ∀ K ∈ N , ∀ P 1 , . . . , P K non commutative polynomials � � P 1 ( a i 1 , a ∗ i 1 ) . . . P K ( a i K , a ∗ τ i K ) = 0 � P k ( a i k , a ∗ � as soon as i 1 � = i 2 � = . . . � = i K and τ i k ) = 0 for k = 1 , . . . , K . Independence vs freeness if a and b are centered ( τ [ a ] = τ [ b ] = 0) free n.c.r.v. then τ [ abab ] = 0, if a and b are independent centered real random variables, E [ abab ] = E [ a 2 ] E [ b 2 ] = 0 iff a or b are non random. Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 10 / 25

Free Probability Voiculescu’s asymptotic freeness Consider X N = ( X ( N ) , . . . , X ( N ) ) be independent N × N GUE matrices p 1 Y N = ( Y ( N ) , . . . , Y ( N ) ) N × N matrices independent with X N . q 1 Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 11 / 25

Free Probability Voiculescu’s asymptotic freeness Consider X N = ( X ( N ) , . . . , X ( N ) ) be independent N × N GUE matrices p 1 Y N = ( Y ( N ) , . . . , Y ( N ) ) N × N matrices independent with X N . q 1 Assumption ∃ n.c.r.v. y = ( y 1 , . . . , y q ) s.t. for Y N viewed as n.c.r.v. in (M k ( C ) , . ∗ , τ N := 1 N Tr ) then when N → ∞ L n . c . i.e. τ N [ P ( Y N , Y ∗ N )] → τ [ P ( y , y ∗ )] ∀ P . − → y Y N Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 11 / 25

Free Probability Voiculescu’s asymptotic freeness Voiculescu (91) Then ∃ n.c.r.v. x = ( x 1 , . . . , x p ) such that ( X N , Y N ) L n . c . i.e. τ N [ P ( X N , Y N , Y ∗ N )] → τ [ P ( x , y , y ∗ )] ∀ P , − → ( x , y ) a.s. and in E when N → ∞ and the law of ( x , y ) is given by x i = x ∗ � i and x i has the semicircular law: τ [ P ( x i ) ] = Pd σ the families ( x 1 , . . . , x p , y ) are free. Camille Mˆ ale (ENS de Lyon) Norm of Polynomials in large RM 12 / 25