A Necessary and Sufficient Condition for Edge Universality of Wigner matrices

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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 $\lim_{s\to \infty}s^4\p(|x_{12}| \ge s)=0$.