Bulk universality for Wigner hermitian matrices with subexponential decay

In this paper, we consider the ensemble of n n Wigner Hermitian matrices H = (h k ) 1 ,kn that generalize the Gaussian unitary ensemble (GUE). The matrix elements h k = h k are given by h k = n -1/2 (x k + -1y k ), where x k , y k for 1 < k n are i.i.d. random variables with mean zero and variance 1/2, y = 0 and x have mean zero and variance 1. We assume the distribution of x k , y k to have subexponential decay. In [3], four of the authors recently established that the gap distribution and averaged k-point correlation of these matrices were universal (and in particular, agreed with those for GUE) assuming additional regularity hypotheses on the x k , y k . In In this short note we observe that the arguments of [3] and [7] can be combined to establish universality of the gap distribution and averaged k-point correlations for all Wigner matrices (with subexponentially decaying entries), with no extra assumptions.