Electronic Journal of Probability

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

Alessandra Faggionato

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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.

Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 73, 2217-2247.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819146

Digital Object Identifier
doi:10.1214/EJP.v13-591

Mathematical Reviews number (MathSciNet)
MR2469609

Zentralblatt MATH identifier
1189.60172

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 82C44: Dynamics of disordered systems (random Ising systems, etc.)

Keywords
disordered system bond percolation random walk in random environment exclusion process homogenization

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Faggionato, Alessandra. Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit. Electron. J. Probab. 13 (2008), paper no. 73, 2217--2247. doi:10.1214/EJP.v13-591. https://projecteuclid.org/euclid.ejp/1464819146


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