Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit
Alessandra Faggionato, Department of Mathematics. University La Sapienza, Rome. Italy
Abstract
We consider a stationary and ergodic random field
parameterized by the family of bonds in Z^d,
d>=2. The random variable associated to the
bond 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 a general criterion developed in a previous paper and
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.
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