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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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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.)

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

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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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