A Necessary and Sufficient Condition for Edge Universality of Wigner matrices

Oon Lee Ji Department of Mathematical Sciences, KAIST. Jun Yin University of Wisconsin

Probability mathscidoc:1608.28007

Duke Mathematical Journal, 163, (1), 117-173, 2012
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.
Edge Universality, Wigner matrice,Tracy-Widom law
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@inproceedings{oon2012a,
  title={A Necessary and Sufficient Condition for Edge Universality of Wigner matrices},
  author={Oon Lee Ji, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823130911693247401},
  booktitle={Duke Mathematical Journal},
  volume={163},
  number={1},
  pages={117-173},
  year={2012},
}
Oon Lee Ji, and Jun Yin. A Necessary and Sufficient Condition for Edge Universality of Wigner matrices. 2012. Vol. 163. In Duke Mathematical Journal. pp.117-173. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823130911693247401.
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