On the universality of the distribution of the eigenvalues of Wigner random matrices in the bulk of the spectrum

TL;DR

This paper proves that the eigenvalue distribution in the bulk of Wigner matrices is universal, matching the Gaussian Orthogonal Ensemble, under mild conditions on entry distributions.

Contribution

It establishes universality of eigenvalue distributions in Wigner matrices near the spectrum center with minimal assumptions on entry distributions.

Findings

Eigenvalue distribution near spectrum center is universal.

Distribution matches Gaussian Orthogonal Ensemble.

Results hold under mild smoothness and decay conditions.

Abstract

In this paper we consider Wigner random matrices -- symmetric n by n random matrices whose entries are independent identically distributed real random variables. We prove that the probability distribution of one or several eigenvalues close to the center of the spectrum does not depend on the probability distribution of the entries of the matrix and is the same as for the Gaussian Orthogonal Ensemble. We make only mild smoothness assumptions on the probability distribution of the entries and assume that the probability distribution of the entries decays polynomially with sufficiently large power or faster than polynomially.