A comment on the Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices

Abstract

Recently we proved that the eigenvalue correlation functions of a general class of random matrices converge, weakly with respect to the energy, to the corresponding ones of Gaussian matrices. Tao and Vu gave a proof that for the special case of Hermitian Wigner matrices the convergence can be strengthened to vague convergence at any fixed energy in the bulk. In this article we comment on this result in the context of the universality conjectures of Mehta. We show that this theorem is an immediate corollary of our earlier results. Indeed, a more general form of this theorem also follows directly from our previous work.