A Necessary and Sufficient Condition for Edge Universality of Wigner matrices

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 $\wt μ$. The means of $\mu$ and $\wt μ$ are zero, the variance of $\mu$ is 1, and the variance of $\wt μ$ 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.

34 pages, 0 figures

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• Local semicircle law and complete delocalization for Wigner random matrices
• Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
• Bulk Universality and Related Properties of Hermitian Matrix Models

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