The Local Circular Law III: General Case

Jun Yin University of Wisconsin

Probability mathscidoc:1608.28008

Distinguished Paper Award in 2017

Probability Theory & Related Fields, 160, (3), 679-732, 2012
In this paper, we prove a necessary and sufficient condition for Tracy-Widom law of Wigner matrices. Consider N×N symmetric Wigner matrices H with Hij=N−1/2xij, whose upper right entries xij (1≤i<j≤N) are i.i.d. random variables with distribution μ and diagonal entries xii (1≤i≤N) are i.i.d. random variables with distribution $\wt \mu$. The means of μ and $\wt \mu$ are zero, the variance of μ 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.
local circular law, universality
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@inproceedings{jun2012the,
  title={The Local Circular Law III: General Case},
  author={Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823134113017204402},
  booktitle={Probability Theory & Related Fields},
  volume={160},
  number={3},
  pages={679-732},
  year={2012},
}
Jun Yin. The Local Circular Law III: General Case. 2012. Vol. 160. In Probability Theory & Related Fields. pp.679-732. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823134113017204402.
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