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

We consider a stationary and ergodic random field that is parameterized by the edge set of the Euclidean lattice , . The random variable , taking values in 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 , 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.