Erdos, LaszloSchlein, BenjaminYau, Horng-Tzer2017-05-182010Erdős, László, Benjamin Schlein, and Horng-Tzer Yau. 2010. “Universality of Random Matrices and Local Relaxation Flow.” Inventiones Mathematicae 185 (1) (December 29): 75–119. doi:10.1007/s00222-010-0302-7.0020-9910http://nrs.harvard.edu/urn-3:HUL.InstRepos:32706727Consider the Dyson Brownian motion with parameter β, where β=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any β≥1, we prove that the relaxation time to local equilibrium for the Dyson Brownian motion is bounded above by N−ζ for some ζ>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 N×N symmetric Wigner ensemble is the same as that of the Gaussian Orthogonal Ensemble (GOE) in the limit N→∞. 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.en-USWigner random matrixDyson Brownian MotionJournal Article2017-05-1810.1007/s00222-010-0302-7