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

Abstract

We consider a stationary and ergodic random field {ω(b) : b ∈ E d } parameterized by the family of bonds in Z d , d2. The random variable ω(b) is thought of as the conductance of bond b and it ranges in a finite interval [0, c0]. Assuming that the set of bonds with positive conductance has a unique infinite cluster C(ω), we prove homogenization results for the random walk among random conductances on C(ω). As a byproduct, applying the general criterion of [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(ω). 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(ω) can be the infinite cluster of supercritical Bernoulli bond percolation.

Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps

“…[DNS18]. Notice that in the nearest-neighbor percolation setting, similar homogenization results have also been obtained by Faggionato in [Fag08] under the additional assumption that the conductances are bounded from above.…”

Homogenization of generalized second-order elliptic difference operators

“…In this Section we recall some notation and results of [5,12,14]. Denote by T d = (R/Z) d = [0, 1) d the d-dimensional torus and fix a function W : R d → R such that (1) W (x 1 , . .…”

“…Recently, the formal adjoint operator (d/dx)(d/dW ) and some non-linear versions were obtained as scaling limits of interacting particle systems in inhomogeneous media. They may model diffusions with permeable membranes at the points of the discontinuities of W , see [1,5,7,2,14] for further details. In [14], for instance, the author introduces an extension of the formal adjoint operator to higher dimensions and provides some results regarding this extension.…”

Homogenization results for a linear dynamics in random Glauber type environment

“…The second term on the left-hand side of the previous inequality is composed by a sum of positive parts. We can restrict this for any m ≤ n. By using (8) and the variational formula (24) for the Dirichlet form, we get that, for any function φ ∈ C 1 0 (Ω) and any functions ψ j ∈ F , j ∈ {−m, . .…”