# Mathematics Colloquia and Seminars

For Gaussian ensembles, the probability distribution of the largest eigenvalue converges to the Tracy-Widom distribution, which is known as the edge universality. In this talk, we discuss a simple criterion for the edge universality result for heavy-tailed Wigner matrices. Consider Wigner matrices $H$ with $H_{ij} = N^{-1/2} x_{ij}$, where the off-diagonal entries are i.i.d. random variables with a distribution $\mu$ and the diagonal entries are i.i.d. random variables with another distribution $\widetilde{\mu}$. If $\mu$ is centered with variance 1 and $\widetilde{\mu}$ is centered with finite variance, then the edge universality holds if and only if $\lim_{s\to \infty}s^4 P(|x_{12}| \geq s)=0$. This is a joint work with Jun Yin.