Hydrodynamic limit of simple exclusion processes in symmetric random environments via duality and homogenization

TL;DR

This paper proves that simple exclusion processes in symmetric random environments exhibit a deterministic hydrodynamic limit under diffusive scaling, with the limit governed by an effective homogenized matrix, covering many complex models.

Contribution

It establishes the hydrodynamic limit for exclusion processes in a broad class of symmetric random environments using duality and homogenization techniques.

Findings

Hydrodynamic limit holds for almost all environments.

Limit governed by the effective homogenized matrix D.

Applicable to various models including conductance models and percolation clusters.

Abstract

We consider continuous-time random walks on a random locally finite subset of $\mathbb{R}^d$ with random symmetric jump probability rates. The jump range can be unbounded. We assume some second--moment conditions and that the above randomness is left invariant by the action of the group $\mathbb{G}=\mathbb{R}^d$ or $\mathbb{G}=\mathbb{Z}^d$. We then add a site-exclusion interaction, thus making the particle system a simple exclusion process. We show that, for almost all environments, under diffusive space-time rescaling the system exhibits a hydrodynamic limit in path space. The hydrodynamic equation is non-random and governed by the effective homogenized matrix $D$ of the single random walk, which can be degenerate. The above result covers a very large family of models including e.g. simple exclusion processes built from random conductance models on $\mathbb{Z}^d$ and on crystal lattices (possibly with long conductances), Mott variable range hopping, simple random walks on Delaunay triangulations, random walks on supercritical percolation clusters.