Hydrodynamic limit of zero range processes among random conductances on the supercritical percolation cluster

We consider i.i.d. random variables {ω(b): b ∈ Ed} parameterized by the family of bonds in Zd, d ≥ 2. The random variable ω(b) is thought of as the conductance of bond b and it ranges in a finite interval [0,c0]. Assuming the probability m of the event {ω(b) > 0} to be supercritical and denoting by C (ω) the unique infinite cluster associated to the bonds with positive conductance, we study the zero range process on C (ω) with ω(b)-proportional probability rate of jumps along bond b. For almost all realizations of the environment we prove that the hydrodynamic behavior of the zero range process is governed by the nonlinear heat equation ∂tρ = m∇ · (D∇φ(ρ/m)), where the matrix D and the function φ are ω-independent. As byproduct of the above result and the blocking effect of the finite clusters, we discuss the bulk behavior of the zero range process on Zd with conductance field ω. We do not require any ellipticity condition.