Universality for generalized Wigner matrices with Bernoulli distribution

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

Probability mathscidoc:1608.28023

Journal of Combinatorics, 2, (1), 15-82, 2011
The universality for the eigenvalue spacing statistics of generalized Wigner matrices was established in our previous work \cite{EYY} under certain conditions on the probability distributions of the matrix elements. A major class of probability measures excluded in \cite{EYY} are the Bernoulli measures. In this paper, we extend the universality result of \cite{EYY} to include the Bernoulli measures so that the only restrictions on the probability distributions of the matrix elements are the subexponential decay and the normalization condition that the variances in each row sum up to one. The new ingredient is a strong local semicircle law which improves the error estimate on the Stieltjes transform of the empirical measure of the eigenvalues from the order $(N \eta)^{-1/2}$ to $(N \eta)^{-1}$. Here $\eta$ is the imaginary part of the spectral parameter in the definition of the Stieltjes transform and $N$ is the size of the matrix.
Random band matrix, local semicircle law
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@inproceedings{lászló2011universality,
  title={Universality for generalized Wigner matrices with Bernoulli distribution},
  author={László Erdos, Horng-Tzer Yau, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160824105615274176431},
  booktitle={Journal of Combinatorics},
  volume={2},
  number={1},
  pages={15-82},
  year={2011},
}
László Erdos, Horng-Tzer Yau, and Jun Yin. Universality for generalized Wigner matrices with Bernoulli distribution. 2011. Vol. 2. In Journal of Combinatorics. pp.15-82. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160824105615274176431.
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