Invariance principle for the random conductance model in a degenerate ergodic environment

TL;DR

This paper proves a quenched invariance principle for a continuous-time random walk in a degenerate, ergodic environment of random conductances, using Moser's iteration to establish sublinearity of the corrector.

Contribution

It establishes a quenched invariance principle for the random conductance model with degenerate conductances under ergodic and moment conditions, advancing understanding of random walks in complex environments.

Findings

Proved a quenched invariance principle for the model.

Established sublinearity of the corrector using Moser's iteration.

Demonstrated the result under specific moment conditions.

Abstract

We study a continuous time random walk, $X$, on ${\mathbb{Z}}^d$ in an environment of random conductances taking values in $(0,\infty)$. We assume that the law of the conductances is ergodic with respect to space shifts. We prove a quenched invariance principle for $X$ under some moment conditions of the environment. The key result on the sublinearity of the corrector is obtained by Moser's iteration scheme.