Eigenvector localization in the heavy-tailed random conductance model

TL;DR

This paper extends previous results on eigenvector localization in heavy-tailed random conductance models to multiple eigenvectors, using spectral analysis techniques to handle sign fluctuations.

Contribution

It introduces a method to analyze the localization of the first k eigenvectors in heavy-tailed conductance models, overcoming challenges posed by fluctuating signs.

Findings

Eigenvectors are localized in heavy-tailed conductance models.

The k-th eigenvector closely approximates an auxiliary spectral problem.

Method extends localization results to multiple eigenvectors.

Abstract

We generalize our former localization result about the principal Dirichlet eigenvector of the i.i.d. heavy-tailed random conductance Laplacian to the first $k$ eigenvectors. We overcome the complication that the higher eigenvectors have fluctuating signs by invoking the Bauer-Fike theorem to show that the $k$th eigenvector is close to the principal eigenvector of an auxiliary spectral problem.