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Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps. (English. French summary) Zbl 1442.60071

Summary: We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in \(\mathbb{Z}^d \). More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum.
We assume that the conductances are stationary, ergodic and nearest-neighbor 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 long-range 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, Poincaré inequalities, Moser iteration and two-scale convergence.

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
60K37 Processes in random environments
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35R60 PDEs with randomness, stochastic partial differential equations
47B80 Random linear operators
47A75 Eigenvalue problems for linear operators
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References:

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