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× 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).
Original language | English (US) |
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Pages (from-to) | 1189-1221 |
Number of pages | 33 |
Journal | Journal of Statistical Physics |
Volume | 174 |
Issue number | 6 |
DOIs | |
State | Published - Mar 30 2019 |
Keywords
- Band random matrix
- Delocalized phase
- Generalized resolvent
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics