Bulk Universality for Wigner Matrices

TL;DR

This paper proves bulk eigenvalue universality for Wigner matrices with certain smoothness and decay conditions on the distribution of entries, using Dyson Brownian motion techniques.

Contribution

It establishes universality for a broad class of Wigner matrices with minimal smoothness assumptions on the distribution.

Findings

Eigenvalue statistics in the bulk follow the Dyson sine kernel

Universality holds under polynomially growing derivatives of the potential

Eigenvalue density converges to the semicircle law on short scales

Abstract

We consider $N\times N$ Hermitian Wigner random matrices $H$ where the probability density for each matrix element is given by the density $\nu(x)= e^{- U(x)}$. We prove that the eigenvalue statistics in the bulk is given by Dyson sine kernel provided that $U \in C^6(\RR)$ with at most polynomially growing derivatives and $\nu(x) \le C e^{- C |x|}$ for $x$ large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales.