Random band matrices in the delocalized phase, II: Generalized resolvent estimates

Paul Bourgade Courant Institute Fan Yang University of California, Los Angeles Horng-Tzer Yau Harvard University Jun Yin University of California, Los Angeles

Mathematical Physics Probability mathscidoc:1808.22001

2018.7
This is the second part of a three part series of papers. In this paper, we consider a general class of $N\times N$ random band matrices $H=(H_{ij})$ whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances $\mathbb E |H_{ij}|^2$ form a band matrix with typical band width $1\ll W\ll N$. We consider the generalized resolvent of $H$ defined as $G(Z):=(H - Z)^{-1}$, where $Z$ is a deterministic diagonal matrix such that $Z_{ij}=\left(z\mathds{1}_{1\le i \le W}+\wt z\mathds{1}_{ i > W} \right) \delta_{ij}$, with two distinct spectral parameters $z\in \mathbb C_+:=\{z\in \mathbb C:\im z>0\}$ and $\wt z\in \mathbb C_+\cup \mathbb R$. In this paper, we prove a sharp bound for the local law of the generalized resolvent $G$ for $W\gg N^{3/4}$. This bound is a key input for the proof of delocalization and bulk universality of random band matrices in [2]. Our proof depends on a fluctuations averaging bound on certain averages of polynomials in the resolvent entries, which will be proved in [10].
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@inproceedings{paul2018random,
  title={Random band matrices in the delocalized phase, II: Generalized resolvent estimates},
  author={Paul Bourgade, Fan Yang, Horng-Tzer Yau, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180822223932548967149},
  year={2018},
}
Paul Bourgade, Fan Yang, Horng-Tzer Yau, and Jun Yin. Random band matrices in the delocalized phase, II: Generalized resolvent estimates. 2018. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180822223932548967149.
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