Bulk Universality for Wigner Hermitian Matrices with Subexponential Decay

We consider the ensemble of n × n Wigner hermitian matrices H = (hlk)1≤l,k≤n that generalize the Gaussian unitary ensemble (GUE). The matrix elements hkl = h̄lk are given by hlk = n (xlk + √ −1ylk), where xlk, ylk for 1 ≤ l < k ≤ n are i.i.d. random variables with mean zero and variance 1/2, yll = 0 and xll have mean zero and variance 1. We assume the distribution of xlk, ylk 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 xlk, ylk. In [7], the other two authors, using a different method, established the same conclusion assuming instead some moment and support conditions on the xlk, ylk. 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.