The Spectral Laws of Hermitian Block-matrices with Large Random Blocks

We are going to study the limiting spectral measure of fixed dimensional Hermitian block-matrices with large dimensional Wigner blocks. We are going also to identify the limiting spectral measure when the Hermitian block-structure is Circulant. Using the limiting spectral measure of a Hermitian Circulant block-matrix we will show that the spectral measure of a Wigner matrix with k−weakly dependent entries need not to be the semicircle law in the limit. 1. Preliminaries and main results Let Mn(C) be the space of all n×n matrices with complex-valued entries. Define the normalized trace of a matrix A = (Aij) n i,j=1 ∈ Mn(C) to be trn(A) := 1 n ∑n i=1 Aii. Definition 1. The spectral measure of a Hermitian n× n matrix A is the probability measure μA given by μA = 1 n n ∑ j=1 δλj where λ1 ≤ λ2 ≤ · · · ≤ λn are the eigenvalues of A and δx is the point mass at x. The weak limit of the spectral measures μAn of a sequence of matrices {An} is called the limiting spectral measure. We will denote the weak convergence of a probability measure μn to μ by μn D −→ μ as n → ∞. Definition 2. A finite symmetric block-structure Bk(a, b, c, . . . ) (or shortly Bk) over a finite alphabet K = {a, b, c, . . .} is a k × k symmetric matrix whose entries are elements in K.