On Eigenvectors of Random Band Matrices with Large Band

TL;DR

This paper investigates the localization and delocalization of eigenvectors in random band matrices, identifying thresholds for different behaviors and revealing properties of eigenvectors at the spectral edge.

Contribution

It establishes new bounds on the band width W where eigenvectors at the spectral edge exhibit specific localization properties, extending understanding beyond previous conjectures.

Findings

Eigenvectors at the spectral edge show mass concentration or strong interaction for W  W  N^{5/7}

Localization occurs for W  N^{1/8} and delocalization for W  N^{4/5}

Transition behaviors at W  N^{1/2} and W  N^{5/6} are partially characterized

Abstract

We study random, symmetric $N \times N$ band matrices with a band of size $W$ and Bernoulli random variables as entries. This interpolates between nearest neighbour interaction $W = 1$ and Wigner matrices $W = N$. Eigenvectors are known to be localized for $W \ll N^{1/8}$, delocalized for $W \gg N^{4/5}$ and it is conjectured that the transition for the bulk occurs at $W \sim N^{1/2}$. Eigenvalues in the spectral edge change their behavior at $W \sim N^{5/6}$ but nothing is known about the associated eigenvectors. We show that up to $W \ll N^{5/7}$ any random matrix has with large probability some eigenvectors in the spectral edge, which either exhibit mass concentration or interact strongly on a small scale.