Flegel, Franziska and Heida, Martin and Slowik, Martin (2019) Homogenization theory for the random conductance model with degenerate ergodic weights and unboundedrange jumps. Ann. Inst. H. Poincaré Probab. Statist., 55 (3). pp. 12261257.

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Official URL: http://doi.org/10.1214/18AIHP917
Abstract
We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Z^d. More precisely, we prove almostsure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearestneighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the longrange connections, the integrability condition on the lower tail is optimal for spectral homogenization. It coincides with a necessary condition for the validity of a local central limit theorem for the random walk among random conductances. As an application of spectral homogenization, we prove a quenched large deviation principle for the normalized and rescaled local times of the random walk in a growing box. Our proofs are based on a compactness result for the Laplacian's Dirichlet energy, Poincare inequalities, Moser iteration and twoscale convergence.
Item Type:  Article 

Subjects:  Mathematical and Computer Sciences > Mathematics > Applied Mathematics 
Divisions:  Department of Mathematics and Computer Science > Institute of Mathematics 
ID Code:  2212 
Deposited By:  Silvia Hoemke 
Deposited On:  15 Feb 2018 15:55 
Last Modified:  17 Jan 2022 14:10 
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