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


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2019-03

AUTHORS

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

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>Wδij, 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). More... »

PAGES

1189-1221

References to SciGraph publications

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s10955-019-02229-z

DOI

http://dx.doi.org/10.1007/s10955-019-02229-z

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1111515447


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