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 $N\times N$ symmetric Wigner matrices $H$ with $H_{ij}=N^{-1/2}{x}_{ij}$ whose upper-right entries $x_{ij}$ $(1\le i<j\le N)$ are independent and identically distributed (i.i.d.) random variables with distribution $\nu$ and diagonal entries $x_{ii}$ $(1\le i\le N)$ are i.i.d. random variables with distribution $\mathrm{\nu ̃}$. The means of $\nu$ and $\mathrm{\nu ̃}$ are zero, the variance of $\nu$ is 1, and the variance of $\mathrm{\nu ̃}$ is finite. We prove that the Tracy–Widom law holds if and only if ${lim\hspace{0.17em}}_{s\to \infty }{s}^4\mathbb{P}\left(\right|x_{12}|\ge s)=0$. The same criterion holds for Hermitian Wigner matrices.