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2018 Localization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model
Franziska Flegel
Electron. J. Probab. 23(none): 1-43 (2018). DOI: 10.1214/18-EJP160

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.

Citation

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Franziska Flegel. "Localization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model." Electron. J. Probab. 23 1 - 43, 2018. https://doi.org/10.1214/18-EJP160

Information

Received: 10 August 2016; Accepted: 21 March 2018; Published: 2018
First available in Project Euclid: 26 July 2018

zbMATH: 06924680
MathSciNet: MR3835474
Digital Object Identifier: 10.1214/18-EJP160

Subjects:
Primary: 47A75, 47B80, 60J27

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