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

Abstract

AbstractWe consider continuous-time random walks on a random locally finite subset of $$\mathbb {R}^d$$ 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 $$\mathbb {G}=\mathbb {R}^d$$ G = R d or $$\mathbb {G}=\mathbb {Z}^d$$ G = 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 $$\mathbb {Z}^d$$ 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.