Laszlo Erdos, Jose A. Ramirez, Benjamin Schlein, Horng-Tzer Yau
Bulk Universality for Wigner Matrices
(81K, latex file)

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.