Abstract:We analyze the rate of convergence of the local statistics of Dyson Brownian motion to the GOE/GUE for short times t " op1q with deterministic initial data V . Our main result states that if the density of states of V is bounded both above and away from 0 down to scales ℓ ! t in a small interval of size G " t around an energy E 0 , then the local statistics coincide with the GOE/GUE near the energy E 0 after time t. Our methods are partly based on the idea of coupling two Dyson Brownian motions from [6], the parabolic regularity result of [15], and the eigenvalue rigidity results of [21].

We consider Dyson Brownian motion for classical values of β with deterministic initial data V . We prove that the local eigenvalue statistics coincide with the GOE/GUE in the fixed energy sense after time t Á 1{N if the density of states of V is bounded above and below down to scales η ! t in a window of size L " ? t. Our results imply that fixed energy universality holds for essentially any random matrix ensemble for which averaged energy universality was previously known. Our methodology builds on the homogenization theory developed in [17] which reduces the microscopic problem to a mesoscopic problem. As an auxiliary result we prove a mesoscopic central limit theorem for linear statistics of various classes of test functions for classical Dyson Brownian motion.

We consider the adjacency matrix of the ensemble of Erdős-Rényi random graphs which consists of graphs on N vertices in which each edge occurs independently with probability p. We prove that in the regime pN ≫ 1, these matrices exhibit bulk universality in the sense that both the averaged n-point correlation functions and distribution of a single eigenvalue gap coincide with those of the GOE. Our methods extend to a class of random matrices which includes sparse ensembles whose entries have different variances. C 2015 AIP Publishing LLC.

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