Eigenvector localization for random band matrices with power law band width

TL;DR

This paper investigates eigenvector localization in random band matrices with power-law band widths, showing that eigenvectors are localized when the band width grows slower than the matrix size, with specific bounds for Gaussian ensembles.

Contribution

It establishes a localization condition for eigenvectors in random band matrices with power-law band widths, extending understanding of eigenvector behavior in such ensembles.

Findings

Eigenvectors strongly overlap with a vanishing fraction of basis vectors.

Localization occurs when band width power $W^$ is less than matrix size $N$.

For Gaussian ensembles, the critical exponent   8.

Abstract

It is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors, provided the band width $W$ raised to a power $\mu$ remains smaller than the matrix size $N$. For a Gaussian band ensemble, with matrix elements given by i.i.d. centered Gaussians within a band of width $W$, the estimate $\mu \le 8$ holds.