Bulk Universality for Wigner Matrices

Laszlo Erdos, Sandrine Peche, Jose A. Ramirez, Benjamin Schlein, Horng-Tzer Yau

Published 2009-05-26, updated 2009-10-20Version 2

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.