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2008 Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit
Alessandra Faggionato
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Electron. J. Probab. 13: 2217-2247 (2008). DOI: 10.1214/EJP.v13-591

Abstract

We consider a stationary and ergodic random field $\{\omega (b):b \in \mathbb{E}_d \}$ parameterized by the family of bonds in $\mathbb{Z}^d$, $d\geq 2$. 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 $\mathcal{C}(\omega)$, we prove homogenization results for the random walk among random conductances on $\mathcal{C}(\omega)$. As a byproduct, applying the general criterion of Faggionato (2007) 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 $\mathcal{C}(\omega)$. 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, $\mathcal{C}(\omega)$ can be the infinite cluster of supercritical Bernoulli bond percolation.

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Alessandra Faggionato. "Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit." Electron. J. Probab. 13 2217 - 2247, 2008. https://doi.org/10.1214/EJP.v13-591

Information

Accepted: 21 December 2008; Published: 2008
First available in Project Euclid: 1 June 2016

zbMATH: 1189.60172
MathSciNet: MR2469609
Digital Object Identifier: 10.1214/EJP.v13-591

Subjects:
Primary: 60K35
Secondary: 60J27, 82C44

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