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

TL;DR

This paper establishes sharp bounds for the generalized resolvent of large random band matrices in the delocalized phase, advancing understanding of their spectral properties and supporting proofs of delocalization and universality.

Contribution

It provides a key local law estimate for the generalized resolvent of band matrices with width W much larger than N^{3/4}, crucial for delocalization proofs.

Findings

Sharp bounds for the generalized resolvent when W≫N^{3/4}

Supports proofs of delocalization and bulk universality

Develops fluctuation averaging bounds for resolvent entries

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 $Z_{ij}=\left(z 1_{1\leq i \leq W}+\widetilde z 1_{ i > W} \right) \delta_{ij}$, with two distinct spectral parameters $z\in \mathbb C_+:=\{z\in \mathbb C:{\rm Im} z>0\}$ 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 \cite{PartI}. Our proof depends on a fluctuations averaging bound on certain averages of polynomials in the resolvent entries, which will be proved in \cite{PartIII}.