Invariance principle for the random conductance model

Abstract

We study a continuous time random walk X in an environment of i.i.d. random conductances $${\mu_{e} \in [0,\infty)}$$ in $${\mathbb{Z}^d}$$ . We assume that $${\mathbb{P}(\mu_{e} > 0) > p_c}$$ , so that the bonds with strictly positive conductances percolate, but make no other assumptions on the law of the μ e . We prove a quenched invariance principle for X, and obtain Green’s functions bounds and an elliptic Harnack inequality.