Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps

Franziska Flegel, Martin Heida, and Martin Slowik

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Abstract

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.

Résumé

Nous considérons les propriétés d’homogénéisation de l’opérateur de Laplace discret avec des conductances aléatoires. Nous démontrons l’homogénéisation de l’équation de Poisson discrète et des plus hauts éléments du spectre de l’opérateur de Dirichlet dans un domaine limité.

Nous supposons les conductances stationnaires, ergodiques et strictement positives à plus proches voisins. Comparé aux résultats précédents, nous remplaçons l’ellipticité uniforme par des conditions d’intégrabilité des moments des conductances. De plus, nous autorisons des sauts de tailles arbitraires.

En l’absence de sauts longs, les conditions sur les moments sont optimales pour l’homogénéisation spectrale. Elles correspondent à la condition nécessaire du théorème central limite pour les marches aléatoires en conductances aléatoires. Nous utilisons l’homogénéisation spectrale pour démontrer un principe de grandes déviations gelé pour le temps local normalisé de la marche aléatoire dans une suite croissante de boites.

Nos démonstrations sont basées sur un résultat de compacité pour l’énergie de Dirichlet, les inégalités de Poincaré, l’itération de Moser et la convergence à deux échelles.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1226-1257.

Dates
Received: 9 February 2017
Revised: 23 March 2018
Accepted: 31 May 2018
First available in Project Euclid: 25 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1569398868

Digital Object Identifier
doi:10.1214/18-AIHP917

Mathematical Reviews number (MathSciNet)
MR4010934

Subjects
Primary: 60H25: Random operators and equations [See also 47B80] 60K37: Processes in random environments 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 47B80: Random operators [See also 47H40, 60H25] 47A75: Eigenvalue problems [See also 47J10, 49R05]

Keywords
Random conductance model Homogenization Dirichlet eigenvalues Local times Percolation

Citation

Flegel, Franziska; Heida, Martin; Slowik, Martin. Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1226--1257. doi:10.1214/18-AIHP917. https://projecteuclid.org/euclid.aihp/1569398868


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