2018/07/04 by Paul Bourgade, Fan Yang, Horng-Tzer Yau +1 · 1 citation
Mathematics · Physics and Astronomy · #math.PR #math-ph #math.MP #msc:15B52 #msc:82B44
paper · pdf · doi:10.1007/s10955-019-02229-z
arxiv created 2018/07/04 · arxiv updated 2019/02/20
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=(Hij) whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances \mathbb E |Hij|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 Zij=(z 11≤ i ≤ W+\widetilde z 1 i > W ) δij, with two distinct spectral parameters z∈ \mathbb C+:=\z∈ \mathbb C:\rm Im z>0\ and \widetilde z∈ \mathbb C+∪ \mathbb R. In this paper, we prove a sharp bound for the local law of the generalized resolvent G for W≫ N3/4. This bound is a key input for the proof of delocalization and bulk universality of random band matrices in \citePartI. Our proof depends on a fluctuations averaging bound on certain averages of polynomials in the resolvent entries, which will be proved in \citePartIII.