Random band matrices in the delocalized phase, III: Averaging fluctuations

Fan Yang Department of Mathematics, University of California, Los Angeles Jun Yin Department of Mathematics, University of California, Los Angeles

Probability mathscidoc:1808.28001

2018.7
We consider a general class of random band matrices $H=(h_{ij})$ whose entries are centered random variables, independent up to a symmetry constraint. We assume the variances $s_{ij}=\mathbb E |h_{ij}|^2$ form a band matrix with typical band width $W\ll N$. Define the {\it{generalized resolvent}} of $H$ as $G(H,Z):=(H - Z)^{-1}$, where $Z$ is a deterministic diagonal matrix with entries $Z_{ii}\in \mathbb C_+$ for all $i$. Then we establish a precise high-probability bound on certain averages of polynomials in the resolvent entries. In particular, compared with the best estimate so far in [5], our result improves the bound by a factor of $W/N$. This fluctuation averaging result is used in [3] to obtain a sharp bound for the local law of the generalized resolvent $G$, which is further used in [4] to prove the delocalization conjecture and bulk universality for random band matrices with $W\gg N^{3/4}$.
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@inproceedings{fan2018random,
  title={Random band matrices in the delocalized phase, III: Averaging fluctuations},
  author={Fan Yang, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180822223335347472148},
  year={2018},
}
Fan Yang, and Jun Yin. Random band matrices in the delocalized phase, III: Averaging fluctuations. 2018. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180822223335347472148.
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