The strong asymptotic freeness of large random and deterministic matrices Camille Male Universit´ e Paris Diderot (Paris 7) Workshop random matrices and their applications, Telecom Paristech, October 8-10 Camille Male (Univ. Pars 7) The strong asymptotic freeness 1 / 24
Introduction Statement of results Camille Male (Univ. Pars 7) The strong asymptotic freeness 2 / 24
Introduction No eigenvalues outside a neighborhood of the lim. support Consider the N by N ′ so called ”separable covariance matrix“ H N , N ′ = A N X N , N ′ B N ′ X ∗ N , N ′ A N , where √ N ′ X N , N ′ : size N × N ′ with i.i.d. standard entries ∼ µ , • • A N , B N ≥ 0: size N × N and N ′ × N ′ resp., s.t. L A N → L a , L B N ′ → L b . Camille Male (Univ. Pars 7) The strong asymptotic freeness 3 / 24
Introduction No eigenvalues outside a neighborhood of the lim. support Consider the N by N ′ so called ”separable covariance matrix“ H N , N ′ = A N X N , N ′ B N ′ X ∗ N , N ′ A N , where √ N ′ X N , N ′ : size N × N ′ with i.i.d. standard entries ∼ µ , • • A N , B N ≥ 0: size N × N and N ′ × N ′ resp., s.t. L A N → L a , L B N ′ → L b . Theorem: Boutet de Mondvel, Khorunzhy and Vasilchuck (96) As N , N ′ → ∞ with c N , N ′ = N N ′ → c > 0, L H N , N ′ → µ ( c ) L a , L b a.s. Theorem: Bai and Silverstein (98), Paul and Silverstein (09) If moreover µ has a finite fourth moment and for N large enough, ( c N , N ′ ) L AN , L BN ⊂ Supp µ ( c ) L a , L b , then, a.s. ∀ ε and for N large enough, Supp µ Sp H N , N ′ ⊂ Supp µ ( c ) L a , L b + ( − ε, ε ) . Camille Male (Univ. Pars 7) The strong asymptotic freeness 3 / 24
Introduction Soft version Theorem : M. (11), Collins, M. (11) X N N × N GUE matrix, U N N × N Haar matrix on U N , Y N = ( Y ( N ) , . . . , Y ( N ) ) arbitrary random N × N matrices, p 1 X N , U N and Y N being independent. Camille Male (Univ. Pars 7) The strong asymptotic freeness 4 / 24
Introduction Soft version Theorem : M. (11), Collins, M. (11) X N N × N GUE matrix, U N N × N Haar matrix on U N , Y N = ( Y ( N ) , . . . , Y ( N ) ) arbitrary random N × N matrices, p 1 X N , U N and Y N being independent. Assume that for any Hermitian matrix H N = P ( Y N , Y ∗ N ), 1 Convergence of the empirical eigenvalues distribution a.s. L H N − N →∞ L h with compact support, → 2 Convergence of the support a.s. for N large enough, Sp H N ⊂ Supp L h + ( − ε, ε ) Then, almost surely, the same properties hold for any Hermitian matrix H N = P ( X N , U N , U ∗ N , Y N , Y ∗ N ) . Camille Male (Univ. Pars 7) The strong asymptotic freeness 4 / 24
Introduction Non commutative probability space Definition : C ∗ -probability space ( A , · ∗ , τ, � · � ) A : C ∗ -algebra, · ∗ : antilinear involution such that ( ab ) ∗ = b ∗ a ∗ ∀ a , b ∈ A , τ : linear form such that τ [ 1 ] = 1, 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. Examples Commutative space: Given a probability space (Ω , F , P ), consider ( L ∞ (Ω , µ ) , ¯ · , E , � · � ∞ ), Matrix spaces: ( M N ( C ) , · ∗ , τ N := 1 N Tr , � · � ). Camille Male (Univ. Pars 7) The strong asymptotic freeness 5 / 24
Introduction Non commutative random variables Proposition If aa ∗ = a ∗ a then there exists a compactly supported probability measure P ( a , a ∗ ) � � � µ a on C such that ∀ P polynomial τ = P ( z , ¯ z ) d µ a ( z ). Moreover � a � = sup {| t | | t ∈ Supp µ a . If A N is an N by N normal matrix, then µ A N = L A N . Camille Male (Univ. Pars 7) The strong asymptotic freeness 6 / 24
Introduction Non commutative random variables Proposition If aa ∗ = a ∗ a then there exists a compactly supported probability measure P ( a , a ∗ ) � � � µ a on C such that ∀ P polynomial τ = P ( z , ¯ z ) d µ a ( z ). Moreover � a � = sup {| t | | t ∈ Supp µ a . If A N is an N by N normal matrix, then µ A N = L A N . Definition � P ( a , a ∗ ) � The map τ a : P �→ τ : law of a = ( a 1 , . . . , a p ). Convergence in n.c. law a N → a : P ( a N , a ∗ P ( a , a ∗ ) � � � � N ) − → , ∀ P , τ N →∞ τ Strong convergence in n.c. law a N → a : CV in n.c. law and � − � P ( a N , a ∗ � � � � P ( a , a ∗ ) � N ) → � , ∀ P . N →∞ Camille Male (Univ. Pars 7) The strong asymptotic freeness 6 / 24
Introduction Interest of this notion for large matrices Let A N = ( A ( N ) , . . . , A ( N ) ) be a family of N by N matrices, and p 1 a = ( a 1 , . . . , a p ) in ( A , . ∗ , τ ). L n . c . N →∞ a N ⇔ ∀ H N = P ( A N , A ∗ Then A N − → N ) Hermitian N →∞ µ h , where h = P ( a N , a ∗ L H N − → N ) . L n . c . N →∞ a N strongly ⇔ ∀ H N = P ( A N , A ∗ Moreover A N − → N ) Hermitian � N →∞ µ h , where h = P ( a N , a ∗ L H N − → N ) , ∀ ε > 0 , ∀ N large , Sp H N ⊂ Supp µ h + ( − ε, ε ) . Camille Male (Univ. Pars 7) The strong asymptotic freeness 7 / 24
Introduction The relation of freeness Definition of freeness The sub-algebras A 1 , . . . , A p are free iff � � � � a j ∈ A i j , i j � = i j +1 , and τ = 0 , ∀ j ≥ 1 ⇒ τ ( a 1 a 2 . . . a n ) = 0 ∀ n ≥ 1 . a j Theorem : Voiculescu X N N × N GUE matrix, U N N × N Haar matrix on U N , Y N = ( Y ( N ) , . . . , Y ( N ) ) arbitrary random N × N matrices, uniformly r 1 bounded, X N , U N and Y N being independent. N →∞ y , then ( X N , U N , Y N ) L n . c . L n . c . If Y N − → N →∞ ( x , u , y ), where x , u and y are free. − → Camille Male (Univ. Pars 7) The strong asymptotic freeness 8 / 24
Introduction The asymptotic freeness of large random matrices Definition : Freeness The sub-algebras A 1 , . . . , A p are free iff � � � � a j ∈ A i j , i j � = i j +1 , and τ = 0 , ∀ j ≥ 1 ⇒ τ ( a 1 a 2 . . . a n ) = 0 ∀ n ≥ 1 . a j Theorem : Voiculescu X N N × N GUE matrix, U N N × N Haar matrix on U N , Y N = ( Y ( N ) , . . . , Y ( N ) ) arbitrary random N × N matrices, uniformly p 1 bounded, X N , U N and Y N being independent. N →∞ y , then ( X N , U N , Y N ) L n . c . L n . c . If Y N − → N →∞ ( x , u , y ), where x , u and y are free. − → Camille Male (Univ. Pars 7) The strong asymptotic freeness 9 / 24
Introduction The strong asymptotic freeness of large random matrices Theorem : Haagerup and Thorbjørnsen, 05 Let X N = ( X ( N ) , . . . , X ( N ) ) be independent GUE matrices. Then p 1 L n . c . N →∞ x strongly, where x = ( x 1 , . . . , x p ) family of free semi-circular − → X N n.c.r.v. Let Y N = ( Y ( N ) , . . . , Y ( N ) ) arbitrary random N × N matrices, such that p 1 L n . c . Y N N →∞ y strongly − → Theorem : M., 11, Collins, M., 11 Let X N be a GUE matrix, U N be a Haar matrix on U N , such that X N , U N and Y N are independent. Then ( X N , U N , Y N ) L n . c . N →∞ ( x , u , y ) strongly, − → where x semi-circular n.c.r.v., u Haar unitary n.c.r.v. and x , u , y are free. Camille Male (Univ. Pars 7) The strong asymptotic freeness 10 / 24
Introduction (Non direct) consequence Proposition: the sum of two Hermitian random matrices, Collins, M. (11) Let A N , B N be two N × N independent Hermitian random matrices. Assume that: 1 the law of one of the matrices is invariant under unitary conjugacy, 2 a.s. L A N − N →∞ L a and L B N − → N →∞ L b compactly supported → 3 a.s. the spectra of the matrices converges to the support of the limiting distribution. Then, a.s. the spectrum of A N + B N converges to the support of µ ⊞ ν , where ⊞ denotes the free additive convolution. Remark: We do not assume that ( A N , B N ) converges strongly ! Camille Male (Univ. Pars 7) The strong asymptotic freeness 11 / 24
Introduction (Non direct) consequence Consider the N by N ′ separable covariance matrix H N , N ′ = A N X N , N ′ B N ′ X ∗ N , N ′ A N , where √ N ′ X N , N ′ is Gaussian, the common distribution µ of the entries of N = α n , N ′ = β n so that c N , N ′ = N N ′ = α β = c . A N and B N converges strongly in n.c. law. Then, a.s. for n large enough, no eigenvalues of H N , N ′ are outside a small neighborhood of the support of the limiting distribution Camille Male (Univ. Pars 7) The strong asymptotic freeness 12 / 24
Proof Idea of the proof Camille Male (Univ. Pars 7) The strong asymptotic freeness 13 / 24
Proof From ( X N , Y N ) to ( U N , Y N ) Based on a coupling ( X N , U N ) between a GUE and a Haar matrix: • Let Z N be a Hermitian matrix. If ( Z N , Y N ) L n . c . N →∞ ( z , y ) strongly and − → f N : R → C CV uniformly to f , then ( f N ( Z N ) , Y N ) L n . c . N →∞ ( f ( z ) , y ) strongly. − → • Let X N = V N ∆ N V ∗ N GUE matrix, F N the cumulative function of its eigenvalues. Then, F N − N →∞ F uniformly and → N = V N Diag ( 1 N , . . . , N H N := F N ( X N ) = V N F N (∆ N ) V ∗ N ) V ∗ N . • Let G − 1 be the inverse cumulative function of the eigenvalues of a Haar N N →∞ G − 1 uniformly and matrix, independent of X N , Y N . Then G − 1 − → N U N := G − 1 N ( H N ) is a Haar matrix. Camille Male (Univ. Pars 7) The strong asymptotic freeness 14 / 24
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