  We consider $N\times N$ symmetric random matrices where the probability
distribution for each matrix element is given by a measure $\nu$ with a
subexponential decay. We prove that the eigenvalue spacing statistics in the
bulk of the spectrum for these matrices and for GOE are the same in the limit
$N \to \infty$. Our approach is based on the study of the Dyson Brownian motion
via a related new dynamics, the local relaxation flow.
