Universality of Random Matrices and Local Relaxation Flow

TL;DR

This paper proves that for large symmetric random matrices with subexponential decay, the local eigenvalue statistics in the bulk match those of GOE, using a novel local relaxation flow approach.

Contribution

It introduces a new local relaxation flow method to establish universality of eigenvalue statistics for a broad class of random matrices.

Findings

Eigenvalue spacing statistics match GOE in the bulk as N approaches infinity.

The approach applies to matrices with subexponential decay in entries.

The method advances understanding of local spectral universality.

Abstract

We consider $N\times N$ symmetric random matrices where the probability distribution for each matrix element is given by a measure $\nu$ with a subexponential decay. We prove that the eigenvalue spacing statistics in the bulk of the spectrum for these matrices and for GOE are the same in the limit $N \to \infty$. Our approach is based on the study of the Dyson Brownian motion via a related new dynamics, the local relaxation flow.