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

DOI10.1214/18-EJP160arXiv1608.02415MaRDI QIDQ6276321

Franziska Flegel

Publication date: 8 August 2016

Abstract: We study the asymptotic behavior of the principal eigenvector and eigenvalue of the random conductance Laplacian in a large domain of $mathbb{Z}^d$ ($dgeq2$) 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 $gamma=supqgeq0colonmathbbE[w^{-q}]<infty<1/4$, 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 $gamm{a}_{mc}=1/4$ 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.

Mathematics Subject Classification ID

Eigenvalue problems for linear operators (47A75) Random linear operators (47B80) Continuous-time Markov processes on discrete state spaces (60J27)

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