Eigenvector localization in the heavy-tailed random conductance model

Flegel, Franziska

doi:10.48550/arxiv.1801.05684

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2018

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“…Note that we do not generalize our results to higher order eigenvectors here since in Lemma 5.6 we rely on the Perron-Frobenius property of the principal Dirichlet eigenvector. In the recent paper [Fle18] we overcome this difficulty by using the Bauer-Fike theorem.…”

Section: Introductionmentioning

confidence: 99%

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

Flegel¹

2018

Electron. J. Probab.

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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 Z d (d ≥ 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 γ = sup{q ≥ 0 : E[w −q ] < ∞} < 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 γ c = 1/4 is sharp. Indeed, other recent results imply that for γ > 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.

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Section: Introductionmentioning

confidence: 99%