Type: Article
Publication Date: 2010-01-01
Citations: 113
DOI: https://doi.org/10.4310/mrl.2010.v17.n4.a7
In this paper, we consider the ensemble of n × n Wigner Hermitian matrices H = (h k ) 1≤ ,k≤n 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 [7], the other two authors, using a different method, established the same conclusion assuming instead some moment and support conditions on the x k , y k .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.