Localization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model

TL;DR

This paper investigates how the principal eigenvector of a random conductance Laplacian localizes in heavy-tailed environments, showing it concentrates at a single site when conductance tails are sufficiently heavy, with a sharp threshold at /4.

Contribution

It establishes the asymptotic localization of the principal eigenvector in heavy-tailed conductance models and identifies a sharp threshold for this behavior.

Findings

Eigenvector concentrates at a single site for /4 > 

Eigenvalue scales subdiffusively in heavy-tailed regimes

Threshold /4 is sharp for localization transition

Abstract

We study the asymptotic behavior of the principal eigenvector and eigenvalue of the random conductance Laplacian in a large domain of $\mathbb{Z}^d$ ($d\geq 2$) with zero Dirichlet condition. We assume that the conductances $w$ are positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. If $\gamma=\sup \{ q\geq 0\colon \mathbb{E} [w^{-q}]<\infty \}<1/4$, then we show that for almost every environment the principal Dirichlet eigenvector asymptotically concentrates in a single site and the corresponding eigenvalue scales subdiffusively. The threshold $\gamma_{\rm c} = 1/4$ is sharp. Indeed, other recent results imply that for $\gamma>1/4$ the top of the Dirichlet spectrum homogenizes. Our proofs are based on a spatial extreme value analysis of the local speed measure, Borel-Cantelli arguments, the Rayleigh-Ritz formula, results from percolation theory, and path arguments.