  We consider a stationary and ergodic random field {\omega(b)} parameterized
by the family of bonds b in Z^d, d>1. 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, we prove homogenization results for the random walk among random
conductances on C. As a byproduct, applying the general criterion of \cite{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. 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 can be the infinite cluster of supercritical
Bernoulli bond percolation.
