The limit spectrum of special random matrices Patryk Pagacz 1 - PowerPoint PPT Presentation

The limit spectrum of special random matrices Patryk Pagacz 1 Department of Mathematics, Jagiellonion University Recent Advances in Operator Theory and Operator Algebras July 13-22, 2016, Bangalore 1 Joint work with Micha� l Wojtylak

Outline Wigner and Marchenko-Pastur theorems Generalized Wigner and Marchenko-Pastur matrices Stochastic domination Isotropic local law for Wigner and Marchenko-Pastur matrices Port-Hamiltonian matrices: Large perturbation of skew-hermitian matrix Deformation of large Wigner matrix Main Theorem

General problem Let A N ∈ C N × N be a sequence of matrices.

General problem Let A N ∈ C N × N be a sequence of matrices. σ ( A N ) → ??,

General problem Let A N ∈ C N × N be a sequence of matrices. σ ( A N ) → ??, where A N is ”almost” hermitian...

Wigner’s theorem Let 1 √ [ x ij ] N W N = ij =0 N stands for the classical Wigner matrix,

Wigner’s theorem Let 1 √ [ x ij ] N W N = ij =0 N stands for the classical Wigner matrix, i.e. W N is symmetric matrix such that x ij are real, E x ij = 0, x ij i.i.d. for i < j (let E | x 01 | 2 = 1), x ii i.i.d., max { E | x 00 | k , E | x 01 | k } < + ∞ k = 1 , 2 , . . .

Wigner’s theorem Let λ N i denote the (real) eigenvalues of a Wigner matrix W N . Let us consider the empirical distribution of the eigenvalues as the (random) probability measure on R defined by N 1 � L N = δ λ N i . N + 1 i =0

Wigner’s theorem Let λ N i denote the (real) eigenvalues of a Wigner matrix W N . Let us consider the empirical distribution of the eigenvalues as the (random) probability measure on R defined by N 1 � L N = δ λ N i . N + 1 i =0 Theorem (Wigner) The empirical measures L N converges weakly, in probability, to the semicircle distribution σ ( x ) dx, where √ σ ( x ) = 1 4 − x 2 χ {| x |≤ 2 } . 2 π

Wigner’s theorem Let λ N i denote the (real) eigenvalues of a Wigner matrix W N . Let us consider the empirical distribution of the eigenvalues as the (random) probability measure on R defined by N 1 � L N = δ λ N i . N + 1 i =0 Theorem (Wigner) The empirical measures L N converges weakly, in probability, to the semicircle distribution σ ( x ) dx, where √ σ ( x ) = 1 4 − x 2 χ {| x |≤ 2 } . 2 π �� � � � � � � fd σ − → 0, for any ε > 0 and f ∈ C b ( R ) i.e. P fdL N � > ε � � �

Wigner’s theorem Figure: An empirical distribution of eigenvalues of Wigner matrix

Marchenko–Pastur law Let X N = ( N ) − 1 2 [ x ij ] ∈ R M × N stands for a matrix such that M / N → y , y ∈ (0 , 1), x ij are i.i.d., E x ij = 0, E | x ij | 2 = 1.

Marchenko–Pastur law Let X N = ( N ) − 1 2 [ x ij ] ∈ R M × N stands for a matrix such that M / N → y , y ∈ (0 , 1), x ij are i.i.d., E x ij = 0, E | x ij | 2 = 1. The matrix X ∗ N X N is called Marchenko-Pastur matrix.

Marchenko–Pastur law denote the (real) eigenvalues of X ∗ Let ν N N X N , the Marchenko-Pastur i matrix. Now let us consider an empirical distribution of ν N i.e. i N 1 � L N = δ ν N i . N + 1 i =0

Marchenko–Pastur law denote the (real) eigenvalues of X ∗ Let ν N N X N , the Marchenko-Pastur i matrix. Now let us consider an empirical distribution of ν N i.e. i N 1 � L N = δ ν N i . N + 1 i =0 Theorem (Marchenko-Pastur) The empirical measures L N converges weakly, in probability, to the Marchenko-Pastur distribution µ with density d µ 1 � dx = ( x − a )( b − x ) χ [ a , b ] , 2 π xy where a = (1 − √ y ) 2 and b = (1 + √ y ) 2 .

Marchenko–Pastur law Figure: An empirical distribution of singular eigenvalues of Marchenko-Pastur matrix

Generalization of Wigner and Marchenko-Pastur matrices Now 1 ij =1 ∈ C N × N √ [ x ij ] N W N = N will stand for the generalized Wigner matrix,

Generalization of Wigner and Marchenko-Pastur matrices Now 1 ij =1 ∈ C N × N √ [ x ij ] N W N = N will stand for the generalized Wigner matrix, i.e. W N is symmetric matrix such that x ij are independent for i ≤ j , E x ij = 0, cosnt ≤ E | x ij | 2 , � j E | x ij | 2 = N , E | x ij | p ≤ const ( p ), for all p ∈ N .

Generalization of Wigner and Marchenko-Pastur matrices Now X N = ( MN ) − 1 4 [ x ij ] ∈ C M × N will stand for a matrix such that 1 const ≤ M ( N ) ≤ N const , N x ij are independent, E x ij = 0, E | x ij | 2 = 1, E | x ij | p = const ( p ), for all p ∈ N .

Generalization of Wigner and Marchenko-Pastur matrices Now X N = ( MN ) − 1 4 [ x ij ] ∈ C M × N will stand for a matrix such that 1 const ≤ M ( N ) ≤ N const , N x ij are independent, E x ij = 0, E | x ij | 2 = 1, E | x ij | p = const ( p ), for all p ∈ N . The matrix X ∗ N X N is called generalized Marchenko-Pastur matrix.

Stochastic domination How to approach the resolvent?

Stochastic domination How to approach the resolvent? In deterministic case we would like to use some inequality � ( W − z ) − 1 − A � ≤ ...

Stochastic domination How to approach the resolvent? In deterministic case we would like to use some inequality � ( W − z ) − 1 − A � ≤ ... We deal with random objects so we need a definition of stochastic domination ≺ instead of ≤ .

Stochastic domination How to approach the resolvent? In deterministic case we would like to use some inequality � ( W − z ) − 1 − A � ≤ ... We deal with random objects so we need a definition of stochastic domination ≺ instead of ≤ . Definition (see [1]) The family of nonnegative random variables ξ = { ξ ( N ) ( z ) : N ∈ N , z ∈ S N } is stochastically dominated in z by ζ = { ζ ( N ) ( z ) : N ∈ N , z ∈ S N } if and only if for all ε > 0 and γ > 0 we have � � � � � ξ ( N ) ( z ) ≤ N ε ζ ( N ) ( z ) ≥ 1 − N − γ , P (1) z ∈ S N for large enough N ≥ N ( ε, γ ).

Stochastic domination Example 1 Let S N = { 0 } , ξ ∼ N (0 , 1) and ζ = log N . Thus for any ε, γ > 0 we N ε log N = N ε ζ with probability greater than 1 − N − γ . have ξ ≤

Stochastic domination Example 1 Let S N = { 0 } , ξ ∼ N (0 , 1) and ζ = log N . Thus for any ε, γ > 0 we N ε log N = N ε ζ with probability greater than 1 − N − γ . have ξ ≤ Figure: An empirical distribution of singular eigenvalues of

Stochastic domination Figure: N ε / log( N ) vs. N (0 , 1)

Stochastic domination Figure: the empirical probability that ξ ≤ N ε ζ

Isotropic local law for Wigner matrix Let us denote by m ( z ) a Stieltjes transform of Wigner semicircle distribution, i.e. √ z 2 − 4 m ( z ) = − z + . 2 Let us consider a family of sets � z = x + i y : | x | ≤ ω − 1 , (log N ) − 1+ ω ≤ y ≤ ω − 1 � S N = , and a family of deterministic functions � Im m ( z ) + 1 Ψ( z ) = Ny . Ny Theorem (A. Knowles, J. Yin) � ( W − z ) − 1 − m ( z ) I � max ≺ Ψ( z )

Isotropic local law for Marchanko-Pastur matrix Let us denote φ + 1 � φ = M / N , γ ± = √ φ ± 2 , K = min( N , M ) . Moreover, let us define the functions � m φ ( z ) = φ 1 / 2 − φ − 1 / 2 − z + i ( z − γ − )( γ + − z ) 2 φ − 1 / 2 z on the sets S N = { z = x + i y ∈ C : (log K ) − 1+ ω ≤ | x | ≤ ω − 1 , (log K ) − 1+ ω ≤ y ≤ ω − 1 , | z | ≥ ω } , and a family of deterministic functions � Im m φ ( z ) + 1 Ψ( z ) = Ny . Ny

Isotropic local law for Marchanko-Pastur matrix Theorem (A. Knowles, J. Yin) � ( X ∗ X − z ) − 1 − m φ ( z ) I � max ≺ Ψ( z )

Nonhermitian case. The main purpose of this talk is to show a limit behavior of eigenvalues of non-hermitian matrices.

Nonhermitian case. The main purpose of this talk is to show a limit behavior of eigenvalues of non-hermitian matrices. In the papers [2, 3] authors showed behavior of a non-real eigenvalue of the matrix HW N , where W N is a Wigner matrix, and H = diag( d , 1 , 1 , . . . , 1), with d < 0.

Port-Hamiltonian: Large perturbation of skew-hermitian matrix Let C ∈ C k × k be a deterministic skew-hermitian matrix, i.e. C = − C ∗ .

Port-Hamiltonian: Large perturbation of skew-hermitian matrix Let C ∈ C k × k be a deterministic skew-hermitian matrix, i.e. C = − C ∗ . And let P = P N ∈ C N × k , Q = Q N ∈ C k × N be the � I k � ∈ C N × k , canonical embeddings, i.e. P N = 0 � � ∈ C k × N . Q N = I k 0

Port-Hamiltonian: Large perturbation of skew-hermitian matrix Let C ∈ C k × k be a deterministic skew-hermitian matrix, i.e. C = − C ∗ . And let P = P N ∈ C N × k , Q = Q N ∈ C k × N be the � I k � ∈ C N × k , canonical embeddings, i.e. P N = 0 � � ∈ C k × N . Q N = I k 0 � C � 0 ∈ C N × N P N CQ N = 0 0

Large perturbation of skew-hermitian matrix How look the non-real eigenvalues of PCQ + X ∗ X ?

Large perturbation of skew-hermitian matrix How look the non-real eigenvalues of PCQ + X ∗ X ? We wonder if PCQ + X ∗ X − z = P ( C − z 2) Q + X ∗ X − z 2 is invertible.