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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 $[0, \infty)$ andsatisfying 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 aunique infinite cluster $\mathcal{C}_{\infty}(\omega)$, we prove a quenchedinvariance principle for the continuous-time random walk among randomconductances under relatively mild conditions on the structure of the infinitecluster. An essential ingredient of our proof is a new anchored relativeisoperimetric inequality.

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