Random matrices: Universality of local eigenvalue statistics

TL;DR

This paper demonstrates that local eigenvalue statistics of random matrices are universal and depend only on the first four moments of entry distributions, applying to various matrix types and statistics.

Contribution

It establishes the universality of local eigenvalue statistics based on the first four moments, covering Wigner Hermitian and real symmetric matrices.

Findings

Eigenvalue gap distribution is universal.

k-point correlation functions are universal.

Results apply under mild assumptions.

Abstract

In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the distribution of the entries. As a consequence, we derive the universality of eigenvalue gap distribution and $k$-point correlation and many other statistics (under some mild assumptions) for both Wigner Hermitian matrices and Wigner real symmetric matrices.