Bulk universality for Wigner matrices

We consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν( x ) = e − U ( x ) . We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U ∈ C 6 ( amssym R ) with at most polynomially growing derivatives and ν( x ) ≥ Ce − 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.