A Necessary and Sufficient Condition for Edge Universality of Wigner matrices

TL;DR

This paper establishes a precise necessary and sufficient condition on the tail behavior of entries for Wigner matrices to exhibit Tracy-Widom edge universality, applicable to both symmetric and Hermitian cases.

Contribution

It provides the first exact criterion linking entry distribution tails to Tracy-Widom law for Wigner matrices, resolving a key question in random matrix theory.

Findings

Tracy-Widom law holds if and only if tail decay condition is met.

The criterion applies to both symmetric and Hermitian Wigner matrices.

The condition involves the asymptotic behavior of the tail probability of matrix entries.

Abstract

In this paper, we prove a necessary and sufficient condition for 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 $i.i.d.$ random variables with distribution $\mu$ and diagonal entries $x_{ii}$ $(1\le i\le N)$ are $i.i.d.$ random variables with distribution $\wt \mu$. The means of $\mu$ and $\wt \mu$ are zero, the variance of $\mu$ is 1, and the variance of $\wt \mu $ is finite. We prove that Tracy-Widom law holds if and only if $\lim_{s\to \infty}s^4\p(|x_{12}| \ge s)=0$. The same criterion holds for Hermitian Wigner matrices.