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

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 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.

The fractional p-Laplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unbounded-range jumps

“…Furthermore, we assume c to be ergodic in Z d×d . This is different from the ergodicity assumptions in [10], where ω x,z is ergodic only in the first variable. The reason is that c x,y = ω x,y |x − y| d+2 in [10] decreases to 0 with growing distance |x − y|, implying E(c) = 0 and causing localization of the operator.…”

“…In a recent work [10], the authors together with Slowik studied homogenization of a discrete Laplace operator on Z d ε := εZ d with long range jumps of the form…”

“…The operator was studied on a bounded domain under proper rescaling with Dirichlet boundary conditions. The coefficients ω x,y being random and positive with ω x,y = ω y,x , the operator L ε acts on functions Z d ε → R, and the corresponding linear equation in [10] reads…”

“…where Q is a bounded open domain in R d . The assumptions on ω x,y imposed in [10] are ergodicity and stationarity in x, together with a first moment condition of the form…”

“…By a non local limit operator, we mean a pseudo differential operator of fractional Laplacian type. Noting that the second order type of the limit operator in [10] is strongly linked to the scaling ε −2 (refer also to [3]), we first relax this scaling to ε −2s , s ∈ (0, 1) obtaining…”