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

Alessandra Faggionato

Published 2020-11-23Version 1

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 homogenized matrix $D$ of the single random walk, which can be degenerate. The above exclusion process is non-gradient but we avoid the standard non-gradient machinery. Our derivation is based on the duality between the simple exclusion process and the random walk and on our recent homogenization results for random walks in random environment presented in \cite{Fhom3}. Our hydrodynamic limit covers a large family of models, including e.g. simple exclusion processes built from random conductance models on $\mathbb{Z}^d$ and on general lattices (possibly with long conductances), Mott variable range hopping, simple random walks on Delaunay triangulations, random walks on supercritical percolation clusters.