Delocalization and Diffusion Profile for Random Band Matrices

TL;DR

This paper investigates the delocalization of eigenvectors and the diffusion behavior of resolvent entries in random band matrices, establishing conditions for delocalization and deriving the diffusion constant in various dimensions.

Contribution

It proves eigenvector delocalization for 1D band matrices with width exceeding L^{4/5} and characterizes the resolvent's behavior via a diffusion operator, extending results to higher dimensions.

Findings

Eigenvectors are delocalized if W  L^{4/5} in 1D.

The resolvent's entries exhibit self-averaging behavior.

The diffusion constant governing the resolvent's behavior is explicitly computed.

Abstract

We consider Hermitian and symmetric random band matrices $H = (h_{xy})$ in $d \geq 1$ dimensions. The matrix entries $h_{xy}$, indexed by $x,y \in (\bZ/L\bZ)^d$, are independent, centred random variables with variances $s_{xy} = \E |h_{xy}|^2$. We assume that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. In one dimension we prove that the eigenvectors of $H$ are delocalized if $W\gg L^{4/5}$. We also show that the magnitude of the matrix entries $\abs{G_{xy}}^2$ of the resolvent $G=G(z)=(H-z)^{-1}$ is self-averaging and we compute $\E \abs{G_{xy}}^2$. We show that, as $L\to\infty$ and $W\gg L^{4/5}$, the behaviour of $\E |G_{xy}|^2$ is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.