Concentration of the empirical spectral distribution of random matrices with dependent entries

Abstract

We investigate concentration properties of spectral measures of Hermitian random matrices with partially dependent entries. More precisely, let $X_{n}$ be a Hermitian random matrix of the size $n\times n$ that can be split into independent blocks of the size at most $d_{n}=o(n^{2})$. We prove that under some mild conditions on the distribution of the entries of $X_{n}$, the empirical spectral measure of $X_{n}$ concentrates around its mean.

The main theorem is a strengthening of the recent result by Kemp and Zimmerman, where the size of the blocks grows as $o(\log n)$. As an application, we are able to upgrade the results of Schenker and Schulz on the convergence in expectation to the semicircle law of a class of random matrices with dependent entries to weak convergence in probability. Other applications include patterned random matrices, e.g. matrices of Toeplitz, Hankel or circulant type and matrices with heavy tailed entries in the domain of attraction of the Gaussian distribution.