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Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit. (English) Zbl 1189.60172
Summary: We consider a stationary and ergodic random field parameterized by the family of bonds in $$\mathbb Z^d, d\geq 2$$. 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.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J27 Continuous-time Markov processes on discrete state spaces 82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics 60K37 Processes in random environments
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