On Eigenvectors of Random Band Matrices with Large Band

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.

This paper has been withdrawn by the author due to a gap in the proof

References in corpus (1)

• Eigenvector localization for random band matrices with power law band width

Cited by in corpus (1)

• Largest eigenvalues and eigenvectors of band or sparse matrices