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

TL;DR

This paper proves that the hydrodynamic limit of zero range processes on supercritical percolation clusters with random conductances follows a nonlinear heat equation, regardless of ellipticity conditions, for almost all environments.

Contribution

It establishes the hydrodynamic limit for zero range processes on random media without ellipticity assumptions, extending previous results to more general environments.

Findings

Hydrodynamic limit follows a nonlinear heat equation.

Results hold for almost all realizations of the environment.

No ellipticity condition required for the proof.

Abstract

We consider i.i.d. random variables {\omega (b):b \in E_d} parameterized by the family of bonds in Z^d, d>1. 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 the probability m of the event {\omega(b)>0} to be supercritical and denoting by C(\omega) the unique infinite cluster associated to the bonds with positive conductance, we study the zero range process on C(\omega) with \omega(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 $\partial_t \rho= m \nabla \cdot (D \nabla\phi(\rho/m))$, where the matrix D and the function \phi are \omega--independent. We do not require any ellipticity condition.