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arXiv · 0704.3020

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

A. Faggionato
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Abstract

We consider a stationary and ergodic random field {ω(b)} parameterized by the family of bonds b in Z^d, d>1. The random variable ω(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.

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A. Faggionato. 2008-09-16. Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit. https://arxiv.org/abs/0704.3020

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