Wigner theorems for random matrices with dependent entries: Ensembles associated to symmetric spaces and sample covariance matrices

TL;DR

This paper extends Wigner's theorem to random matrices from classical symmetric spaces and sample covariance matrices with dependent entries, demonstrating universality of eigenvalue distributions under broad conditions.

Contribution

It introduces new universality results for eigenvalue distributions of matrices from symmetric spaces and dependent sample covariance matrices, generalizing classical Wigner results.

Findings

Eigenvalue distributions converge to universal laws

Results hold under dependence among entries

Includes models from condensed matter physics

Abstract

It is a classical result of Wigner that for an hermitian matrix with independent entries on and above the diagonal, the mean empirical eigenvalue distribution converges weakly to the semicircle law as matrix size tends to infinity. In this paper, we prove analogs of Wigner's theorem for random matrices taken from all infinitesimal versions of classical symmetric spaces. This is a class of models which contains those studied by Wigner and Dyson, along with seven others arising in condensed matter physics. Like Wigner's, our results are universal in that they only depend on certain assumptions about the moments of the matrix entries, but not on the specifics of their distributions. What is more, we allow for a certain amount of dependence among the matrix entries, in the spirit of a recent generalization of Wigner's theorem, due to Schenker and Schulz-Baldes. As a byproduct, we obtain a universality result for sample covariance matrices with dependent entries.