Universality of random matrices with correlated entries

Abstract

We consider an $N$ by $N$ real symmetric random matrix $X=(x_{ij})$ where $\mathbb{E} [x_{ij}x_{kl}]=\xi _{ijkl}$. Under the assumption that $(\xi _{ijkl})$ is the discretization of a piecewise Lipschitz function and that the correlation is short-ranged we prove that the empirical spectral measure of $X$ converges to a probability measure. The Stieltjes transform of the limiting measure can be obtained by solving a functional equation. Under the slightly stronger assumption that $(x_{ij})$ has a strictly positive definite covariance matrix, we prove a local law for the empirical measure down to the optimal scale $\operatorname{Im} z \gtrsim N^{-1}$. The local law implies delocalization of eigenvectors. As another consequence we prove that the eigenvalue statistics in the bulk agrees with that of the GOE.