Averaging Fluctuations in Resolvents of Random Band Matrices
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].
Keywords
Cite
@article{arxiv.1205.5664,
title = {Averaging Fluctuations in Resolvents of Random Band Matrices},
author = {Laszlo Erdos and Antti Knowles and Horng-Tzer Yau},
journal= {arXiv preprint arXiv:1205.5664},
year = {2015}
}