Electronic Journal of Probability

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

Franziska Flegel

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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.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 68, 43 pp.

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

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Zentralblatt MATH identifier

Primary: 47B80: Random operators [See also 47H40, 60H25] 47A75: Eigenvalue problems [See also 47J10, 49R05] 60J27: Continuous-time Markov processes on discrete state spaces

random conductance model Dirichlet spectrum random walk extreme value analysis path arguments percolation Borel-Cantelli variational formula

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Flegel, Franziska. Localization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model. Electron. J. Probab. 23 (2018), paper no. 68, 43 pp. doi:10.1214/18-EJP160. https://projecteuclid.org/euclid.ejp/1532570596

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