Universality of random matrices with correlated entries

TL;DR

This paper proves that the spectral distribution of large correlated random matrices converges to a universal limit, with local laws and eigenvector delocalization, extending results known for independent entries.

Contribution

It establishes universality results for correlated random matrices with short-range dependencies, including local spectral laws and eigenvector delocalization.

Findings

Spectral measure converges to a universal limit.

Local law holds at optimal scale.

Eigenvectors are delocalized, and bulk eigenvalue statistics match GOE.

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 $\text{Im} z \gtrsim N^{-1}$. 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.