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

P. Bourgade, F. Yang, H. T. Yau, J. Yin

Research output: Contribution to journalArticlepeer-review

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 languageEnglish (US)
Pages (from-to)1189-1221
Number of pages33
JournalJournal of Statistical Physics
Volume174
Issue number6
DOIs
StatePublished - Mar 30 2019

Keywords

  • Band random matrix
  • Delocalized phase
  • Generalized resolvent

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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