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

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× N random band matrices H= (H _ij ) whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances E| H _ij | ² form a band matrix with typical band width 1 ≪ W≪ 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 = (z1 _{1 ⩽ i ⩽ W} + z~ 1 _{i > W} ) δ _ij , with two distinct spectral parameters z∈C+:={z∈C:Imz>0} and z~ ∈ C ₊ ∪ R. In this paper, we prove a sharp bound for the local law of the generalized resolvent G for W≫ 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).