Quenched invariance principles for the random conductance model on a random graph with degenerate ergodic weights

Jean-Dominique Deuschel, Tuan Anh Nguyen, Martin Slowik

Published 2016-02-26Version 1

We consider a stationary and ergodic random field $\{\omega(e) : e \in E_d\}$ that is parameterized by the edge set of the Euclidean lattice $\mathbb{Z}^d$, $d \geq 2$. The random variable $\omega(e)$, taking values in $[0, \infty)$ and satisfying certain moment bounds, is thought of as the conductance of the edge $e$. Assuming that the set of edges with positive conductances give rise to a unique infinite cluster $\mathcal{C}_{\infty}(\omega)$, we prove a quenched invariance principle for the continuous-time random walk among random conductances under certain moment conditions. An essential ingredient of our proof is a new anchored relative isoperimetric inequality.