A NECESSARY AND SUFFICIENT CONDITION FOR EDGE UNIVERSALITY OF WIGNER MATRICES

Abstract

In this paper, we prove a necessary and sufficient condition for the Tracy Widom law of Wigner matrices. Consider N x N symmetric Wigner matrices H with H(i)j N-(1/2)x(ij) whose upper-right entries xisi (1 <= i <f <= N) are independent and identically distributed (i.i.d.) random variables with distribution v and diagonal entries xii (1 <= i <= N) are i.i.d. random variables with distribution 13. The means of v and 15 are zero, the variance of v is 1, and the variance of 13 is finite. We prove that the Tracy Widom law holds if and only if lim(s ->infinity) s(4) P(vertical bar x(12)vertical bar >= s) = 0. The same criterion holds for Hermitian Wigner matrices.