Quantum diffusion and delocalization in one-dimensional band matrices via the flow method

TL;DR

This paper investigates quantum diffusion and eigenvector delocalization in one-dimensional Gaussian band matrices, establishing conditions under which these phenomena occur using a flow method, with potential for improved bounds.

Contribution

It introduces a flow method to analyze quantum diffusion and delocalization in Gaussian band matrices, providing new bounds on the band-width requirement.

Findings

Delocalization holds for bulk eigenvectors when W ≫ N^{8/11}.

Quantum diffusion is demonstrated for the resolvent under the same condition.

The flow method is refined, suggesting possible improvements on the band-width condition.

Abstract

We study a class of Gaussian random band matrices of dimension $N \times N$ and band-width $W$. We show that delocalization holds for bulk eigenvectors and that quantum diffusion holds for the resolvent, all under the assumption that $W \gg N^{8/11}$. Our analysis is based on a flow method, and a refinement of it may lead to an improvement on the condition $W \gg N^{8/11}$.