Moments of Traces for Circular -ensembles Tiefeng Jiang University - PowerPoint PPT Presentation

Moments of Traces for Circular β -ensembles Tiefeng Jiang University of Minnesota This is a joint work with Sho Matsumoto September 19, 2014

Outline Moments for Haar Unitary Matrices (D.E. Thm) Background for Circular β -Ensembles Moments for Circular β -Ensembles Proof by Jack Polynomials CLT

1. Moments for Haar Unitary Matrices ◮ What is Haar-invariant unitary matrix Γ n ? Mathematically, Γ n : Haar probability measure on U ( n ) : set of n by n unitary matrices.

1. Moments for Haar Unitary Matrices ◮ What is Haar-invariant unitary matrix Γ n ? Mathematically, Γ n : Haar probability measure on U ( n ) : set of n by n unitary matrices. Statistically, Assume the entries of Y = Y n × n are i.i.d. C N ( 0 , 1 ) . Two ways to generate such matrices

1. Moments for Haar Unitary Matrices ◮ What is Haar-invariant unitary matrix Γ n ? Mathematically, Γ n : Haar probability measure on U ( n ) : set of n by n unitary matrices. Statistically, Assume the entries of Y = Y n × n are i.i.d. C N ( 0 , 1 ) . Two ways to generate such matrices 1) The matrix Q in QR (Gram-Schmidt) decomposition of Y

1. Moments for Haar Unitary Matrices ◮ What is Haar-invariant unitary matrix Γ n ? Mathematically, Γ n : Haar probability measure on U ( n ) : set of n by n unitary matrices. Statistically, Assume the entries of Y = Y n × n are i.i.d. C N ( 0 , 1 ) . Two ways to generate such matrices 1) The matrix Q in QR (Gram-Schmidt) decomposition of Y d = Y ( Y ∗ Y ) − 1 / 2 2) Γ n

◮ Theorem (Diaconis and Evans: 2001) (a) a = ( a 1 , · · · , a k ) , b = ( b 1 , · · · , b k ) with a j , b j ∈ { 0 , 1 , 2 , · · · } . For n ≥ � k j = 1 ja j ∨ � k j = 1 jb j ,

◮ Theorem (Diaconis and Evans: 2001) (a) a = ( a 1 , · · · , a k ) , b = ( b 1 , · · · , b k ) with a j , b j ∈ { 0 , 1 , 2 , · · · } . For n ≥ � k j = 1 ja j ∨ � k j = 1 jb j ,   k k � ( Tr ( U j n )) a j ( Tr ( U j  = δ ab � j a j a j ! n )) b j E  j = 1 j = 1

◮ Theorem (Diaconis and Evans: 2001) (a) a = ( a 1 , · · · , a k ) , b = ( b 1 , · · · , b k ) with a j , b j ∈ { 0 , 1 , 2 , · · · } . For n ≥ � k j = 1 ja j ∨ � k j = 1 jb j ,   k k � ( Tr ( U j n )) a j ( Tr ( U j  = δ ab � j a j a j ! n )) b j E  j = 1 j = 1 (b) For j and k , Tr ( U j � � n ) Tr ( U k n ) = δ jk · j ∧ n . E

O(n) COE Classical Circular Compact U(n)=CUE Ensembles Groups CSE Sp(n) Circular Ensembles and Haar-invariant Matrices from Classical Compact Groups

O(n) COE Classical Circular Compact U(n)=CUE Ensembles Groups CSE Sp(n) Circular Ensembles and Haar-invariant Matrices from Classical Compact Groups Diaconis (2004) believes there is a good formula for COE and CSE

2. Background for Circular β -Ensembles ◮ Probability density function e i θ 1 , · · · , e i θ n : eigenvalues of Haar-invariant unitary matrix. pdf: f ( θ 1 , · · · , θ n | β = 2 )

2. Background for Circular β -Ensembles ◮ Probability density function e i θ 1 , · · · , e i θ n : eigenvalues of Haar-invariant unitary matrix. pdf: f ( θ 1 , · · · , θ n | β = 2 ) | e i θ j − e i θ k | β � f ( θ 1 , · · · , θ n | β ) = Const · 1 ≤ j < k ≤ n β > 0, θ i ∈ [ 0 , 2 π )

2. Background for Circular β -Ensembles ◮ Probability density function e i θ 1 , · · · , e i θ n : eigenvalues of Haar-invariant unitary matrix. pdf: f ( θ 1 , · · · , θ n | β = 2 ) | e i θ j − e i θ k | β � f ( θ 1 , · · · , θ n | β ) = Const · 1 ≤ j < k ≤ n β > 0, θ i ∈ [ 0 , 2 π ) This model: circular β -ensemble by physicist Dyson for study of nuclear scattering data. Matrix model by Killip & Nenciu (04)

◮ Three Important Circular Ensembles COE ( β = 1), CUE ( β = 2), CSE ( β = 4)

◮ Three Important Circular Ensembles COE ( β = 1), CUE ( β = 2), CSE ( β = 4) Construction of COE and CUE U = U n × n : Haar unitary

◮ Three Important Circular Ensembles COE ( β = 1), CUE ( β = 2), CSE ( β = 4) Construction of COE and CUE U = U n × n : Haar unitary U follows CUE

◮ Three Important Circular Ensembles COE ( β = 1), CUE ( β = 2), CSE ( β = 4) Construction of COE and CUE U = U n × n : Haar unitary U follows CUE U T U follows COE

◮ Three Important Circular Ensembles COE ( β = 1), CUE ( β = 2), CSE ( β = 4) Construction of COE and CUE U = U n × n : Haar unitary U follows CUE U T U follows COE CSE is similar but a bit involved (see Mehta)

◮ Three Important Circular Ensembles COE ( β = 1), CUE ( β = 2), CSE ( β = 4) Construction of COE and CUE U = U n × n : Haar unitary U follows CUE U T U follows COE CSE is similar but a bit involved (see Mehta) Entries of CUE : roughly independent C N ( 0 , 1 ) (Jiang, AP06)

◮ Three Important Circular Ensembles COE ( β = 1), CUE ( β = 2), CSE ( β = 4) Construction of COE and CUE U = U n × n : Haar unitary U follows CUE U T U follows COE CSE is similar but a bit involved (see Mehta) Entries of CUE : roughly independent C N ( 0 , 1 ) (Jiang, AP06) Entries of COE : roughly C N ( 0 , 1 ) (but dependent) (Jiang, JMP09)

Moments for Circular β -Ensembles

Moments for Circular β -Ensembles ◮ Bad news from COE: Let M n be COE. By elementary check 2 n | Tr ( M n ) | 2 � � E = n + 1

Moments for Circular β -Ensembles ◮ Bad news from COE: Let M n be COE. By elementary check 2 n | Tr ( M n ) | 2 � � E = n + 1 Moments depend on n

Moments for Circular β -Ensembles ◮ Bad news from COE: Let M n be COE. By elementary check 2 n | Tr ( M n ) | 2 � � E = n + 1 Moments depend on n � | Tr ( M n ) | 2 � Later results: E not depend on n only at β = 2

Moments for Circular β -Ensembles ◮ Bad news from COE: Let M n be COE. By elementary check 2 n | Tr ( M n ) | 2 � � E = n + 1 Moments depend on n � | Tr ( M n ) | 2 � Later results: E not depend on n only at β = 2 This suggest: moments for general β -ensemble depend on n

◮ Notation λ = ( λ 1 , λ 2 , · · · ) : partition

◮ Notation λ = ( λ 1 , λ 2 , · · · ) : partition | λ | = λ 1 + λ 2 + · · · : weight

◮ Notation λ = ( λ 1 , λ 2 , · · · ) : partition | λ | = λ 1 + λ 2 + · · · : weight m i ( λ ) : multi of i in ( λ 1 , λ 2 , · · · )

◮ Notation λ = ( λ 1 , λ 2 , · · · ) : partition | λ | = λ 1 + λ 2 + · · · : weight m i ( λ ) : multi of i in ( λ 1 , λ 2 , · · · ) l ( λ ) = # of positive λ i in λ : length

◮ Notation λ = ( λ 1 , λ 2 , · · · ) : partition | λ | = λ 1 + λ 2 + · · · : weight m i ( λ ) : multi of i in ( λ 1 , λ 2 , · · · ) l ( λ ) = # of positive λ i in λ : length � i m i ( λ ) m i ( λ )! z λ = i ≥ 1

◮ Notation λ = ( λ 1 , λ 2 , · · · ) : partition | λ | = λ 1 + λ 2 + · · · : weight m i ( λ ) : multi of i in ( λ 1 , λ 2 , · · · ) l ( λ ) = # of positive λ i in λ : length � i m i ( λ ) m i ( λ )! z λ = i ≥ 1 p λ = � l ( λ ) i = 1 p λ i , where p k ( x 1 , x 2 , · · · ) = x k 1 + x k 2 + · · ·

◮ Notation λ = ( λ 1 , λ 2 , · · · ) : partition | λ | = λ 1 + λ 2 + · · · : weight m i ( λ ) : multi of i in ( λ 1 , λ 2 , · · · ) l ( λ ) = # of positive λ i in λ : length � i m i ( λ ) m i ( λ )! z λ = i ≥ 1 p λ = � l ( λ ) i = 1 p λ i , where p k ( x 1 , x 2 , · · · ) = x k 1 + x k 2 + · · · λ = ( 3 , 2 , 2 ) : | λ | = 7 , m 2 ( λ ) = 2 , m 3 ( λ ) = 1, l ( λ ) = 3 , i λ 3 i λ 2 i ) 2 p λ = ( � i ) · ( �

α > 0, K ≥ 1, n ≥ 1 , define | α − 1 | � K � 1 − n − K + α δ ( α ≥ 1 ) A = | α − 1 | � K � B = 1 + n − K + α δ ( α < 1 )

α > 0, K ≥ 1, n ≥ 1 , define | α − 1 | � K � 1 − n − K + α δ ( α ≥ 1 ) A = | α − 1 | � K � B = 1 + n − K + α δ ( α < 1 ) Let θ 1 , · · · , θ n ∼ f ( θ 1 , · · · , θ n | β ) , α = 2 /β. Z n = ( e i θ 1 , · · · , e i θ n ) , p µ ( Z n ) = p µ ( e i θ 1 , · · · , e i θ n )

Theorem (a) If n ≥ K = | µ | , then � | p µ ( Z n ) | 2 � A ≤ E ≤ B α l ( µ ) z µ

Theorem (a) If n ≥ K = | µ | , then � | p µ ( Z n ) | 2 � A ≤ E ≤ B α l ( µ ) z µ � � (b) If | µ | � = | ν | , then E p µ ( Z n ) p ν ( Z n ) = 0 .

Theorem (a) If n ≥ K = | µ | , then � | p µ ( Z n ) | 2 � A ≤ E ≤ B α l ( µ ) z µ � � (b) If | µ | � = | ν | , then E p µ ( Z n ) p ν ( Z n ) = 0 . If µ � = ν and n ≥ K = | µ | ∨ | ν | , then � �� � � � � ≤ max {| A − 1 | , | B − 1 |}· α ( l ( µ )+ l ( ν )) / 2 ( z µ z ν ) 1 / 2 p µ ( Z n ) p ν ( Z n ) � E � � � �

Theorem (a) If n ≥ K = | µ | , then � | p µ ( Z n ) | 2 � A ≤ E ≤ B α l ( µ ) z µ � � (b) If | µ | � = | ν | , then E p µ ( Z n ) p ν ( Z n ) = 0 . If µ � = ν and n ≥ K = | µ | ∨ | ν | , then � �� � � � � ≤ max {| A − 1 | , | B − 1 |}· α ( l ( µ )+ l ( ν )) / 2 ( z µ z ν ) 1 / 2 p µ ( Z n ) p ν ( Z n ) � E � � � � (c) ∃ C = C ( β ) s.t. ∀ m ≥ 1 , n ≥ 2 � ≤ Cn 3 2 n β � � | p m ( Z n ) | 2 � � − n � E � � m 1 ∧ β