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

  • P. Bourgade
  • F. Yang
  • H.-T. Yau
  • J. YinEmail author
Article
  • 15 Downloads

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 \(N\times N\) random band matrices \(H=(H_{ij})\) whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances \(\mathbb {E} |H_{ij}|^2\) form a band matrix with typical band width \(1\ll W\ll N\). We consider the generalized resolvent of H defined as \(G(Z):=(H - Z)^{-1}\), where Z is a deterministic diagonal matrix such that \(Z_{ij}=\left( z\mathbb {1}_{1\leqslant i \leqslant W}+\widetilde{z}\mathbb {1}_{ i > W} \right) \delta _{ij}\), with two distinct spectral parameters \(z\in \mathbb {C}_+:=\{z\in \mathbb {C}:{{\,\mathrm{Im}\,}}z>0\}\) and \(\widetilde{z}\in \mathbb {C}_+\cup \mathbb {R}\). In this paper, we prove a sharp bound for the local law of the generalized resolvent G for \(W\gg N^{3/4}\). 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).

Keywords

Band random matrix Delocalized phase Generalized resolvent 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Courant InstituteNew YorkUSA
  2. 2.University of California, Los AngelesLos AngelesUSA
  3. 3.Harvard UniversityCambridgeUSA

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