Universality of Random Matrices and Local Relaxation Flow

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We consider N× N symmetric random matrices where the probability distribution for each matrix element is given by a measure ν 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 → ∞. Our approach is based on the study of the Dyson Brownian motion via a related new dynamics, the local relaxation flow.