Averaging Fluctuations in Resolvents of Random Band Matrices

TL;DR

This paper establishes high-probability bounds on averages of monomials in resolvent entries of random band matrices, advancing understanding of their spectral properties and eigenvector delocalization.

Contribution

It generalizes previous bounds to higher order monomials for random band matrices, enabling proofs of diffusion approximation and eigenvector delocalization.

Findings

Bounds apply to random band matrices with improved order from 2 to 4

Proves diffusion approximation for resolvent magnitude

Derives new delocalization bounds for eigenvectors

Abstract

We consider a general class of random matrices whose entries are centred random variables, independent up to a symmetry constraint. We establish precise high-probability bounds on the averages of arbitrary monomials in the resolvent matrix entries. Our results generalize the previous results of [5,16,17] which constituted a key step in the proof of the local semicircle law with optimal error bound in mean-field random matrix models. Our bounds apply to random band matrices, and improve previous estimates from order 2 to order 4 in the cases relevant for applications. In particular, they lead to a proof of the diffusion approximation for the magnitude of the resolvent of random band matrices. This, in turn, implies new delocalization bounds on the eigenvectors. The applications are presented in a separate paper [3].