Universality of random matrices and local relaxation flow

Consider the Dyson Brownian motion with parameter beta, where beta=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any beta a parts per thousand yen1, we prove that the relaxation time to local equilibrium for the Dyson Brownian motion is bounded above by N (-zeta) for some zeta > 0. The proof is based on an estimate of the entropy flow of the Dyson Brownian motion w.r.t. a pseudo equilibrium measure. As an application of this estimate, we prove that the eigenvalue spacing statistics in the bulk of the spectrum for NxN symmetric Wigner ensemble is the same as that of the Gaussian Orthogonal Ensemble (GOE) in the limit N -> a. The assumptions on the probability distribution of the matrix elements of the Wigner ensemble are a subexponential decay and some minor restriction on the support.