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

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

Research output: Contribution to journalArticle

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= (Hij) whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances 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= (z11 i W+ z~ 1i > W) δij, 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≫ N3 / 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)
JournalJournal of Statistical Physics
DOIs
StateAccepted/In press - Jan 1 2019

Fingerprint

Resolvent Estimates
Band matrix
Random Matrices
Resolvent
estimates
matrices
entry
random variables
Sharp Bound
Diagonal matrix
Independent Random Variables
Universality
Averaging
polynomials
Bandwidth
Fluctuations
bandwidth
Distinct
Symmetry
Polynomial

Keywords

  • Band random matrix
  • Delocalized phase
  • Generalized resolvent

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Random Band Matrices in the Delocalized Phase, II : Generalized Resolvent Estimates. / Bourgade, Paul; Yang, F.; Yau, H. T.; Yin, J.

In: Journal of Statistical Physics, 01.01.2019.

Research output: Contribution to journalArticle

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