Bulk universality for generalized Wigner matrices

Laszlo Erdos University of Munich Horng-Tzer Yau Department of Mathematics, Harvard University Jun Yin Department of Mathematics, Harvard University

Probability mathscidoc:1608.28018

Probability Theory & Related Fields, 154, (1), 1-67, 2010
Consider $N\times N$ Hermitian or symmetric random matrices $H$ where the distribution of the $(i,j)$ matrix element is given by a probability measure $\nu_{ij}$ with a subexponential decay. Let $\sigma_{ij}^2$ be the variance for the probability measure $\nu_{ij}$ with the normalization property that $\sum_{i} \sigma^2_{ij} = 1$ for all $j$. Under essentially the only condition that $c\le N \sigma_{ij}^2 \le c^{-1}$ for some constant $c>0$, we prove that, in the limit $N \to \infty$, the eigenvalue spacing statistics of $H$ in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth $M$ the local semicircle law holds to the energy scale $M^{-1}$.
Random band matrix, Local semicircle law, sine kernel
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  title={Bulk universality for generalized Wigner matrices},
  author={Laszlo Erdos, Horng-Tzer Yau, and Jun Yin},
  booktitle={Probability Theory & Related Fields},
Laszlo Erdos, Horng-Tzer Yau, and Jun Yin. Bulk universality for generalized Wigner matrices. 2010. Vol. 154. In Probability Theory & Related Fields. pp.1-67. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160824093448452690426.
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