WIAS Preprint No. 2371, (2017)

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


  • Flegel, Franziska
  • Heida, Martin
    ORCID: 0000-0002-7242-8175
  • Slowik, Martin

2010 Mathematics Subject Classification

  • 60H25 60K37 35B27 35R60 47B80 47A75


  • Random conductance model, homogenization, Dirichlet eigenvalues, local times, percolation




We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Zd. 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 thenormalized 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

Appeared in

  • Ann. Inst. H. Poincare Probab. Statist., 55 (2019), pp. 1226--1257, DOI 10.1214/18-AIHP917 .

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