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

## TL;DR

This paper establishes homogenization results for the discrete Laplace operator with random, possibly degenerate conductances and long-range jumps, extending previous work by relaxing uniform ellipticity assumptions.

## Contribution

It introduces homogenization theory for models with degenerate ergodic weights and unbounded jumps, providing new conditions for spectral convergence and large deviation principles.

## Key findings

- Almost-sure homogenization of the discrete Poisson equation
- Spectral homogenization under integrability conditions
- Quenched large deviation principle for local times

## 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, Poincare inequalities, Moser iteration and two-scale convergence.