Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit

We consider a stationary and ergodic random field {omega(b) : b epsilon E-d} parameterized by the family of bonds in Z(d), d >= 2. The random variable omega(b) is thought of as the conductance of bond b and it ranges in a finite interval [0, c(0)]. Assuming that the set of bonds with positive conductance has a unique infinite cluster C (omega), we prove homogenization results for the random walk among random conductances on C(omega). As a byproduct, applying the general criterion of [F] leading to the hydrodynamic limit of exclusion processes with bond-dependent transition rates, for almost all realizations of the environment we prove the hydrodynamic limit of simple exclusion processes among random conductances on C(omega). The hydrodynamic equation is given by a heat equation whose diffusion matrix does not depend on the environment. We do not require any ellipticity condition. As special case, C(omega) can be the infinite cluster of supercritical Bernoulli bond percolation.