Random Band Matrices in the Delocalized Phase, II: Generalized Resolvent Estimates

Abstract

This is the second part of a three part series abut delocalization for band matrices. 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 $Z_{ij}=\left( z\mathbb {1}_{1\leqslant i \leqslant W}+\widetilde{z}\mathbb {1}_{ i > W} \right) \delta _{ij}$, with two distinct spectral parameters $z\in \mathbb {C}_+:=\{z\in \mathbb {C}:{{\,\mathrm{Im}\,}}z>0\}$ and $\widetilde{z}\in \mathbb {C}_+\cup \mathbb {R}$. 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 Bourgade et al. (arXiv:1807.01559, 2018). Our proof depends on a fluctuations averaging bound on certain averages of polynomials in the resolvent entries, which will be proved in Yang and Yin (arXiv:1807.02447, 2018).