Hydrodynamic limit for a system of independent, sub-ballistic random walks in a common random environment

TL;DR

This paper establishes a hydrodynamic limit for a system of independent, sub-ballistic random walks in a common random environment, revealing environment-dependent asymptotic particle densities under specific scaling.

Contribution

It extends hydrodynamic limit results to sublinear speed regimes where environment effects persist, using a novel approach with directed trap environments.

Findings

Hydrodynamic limit depends on the specific environment, not averaging out.

Scaling involves time by n^{1/κ} and space by n.

Environment influences the asymptotic particle density.

Abstract

We consider a system of independent random walks in a common random environment. Previously, a hydrodynamic limit for the system of RWRE was proved under the assumption that the random walks were transient with positive speed. In this paper we instead consider the case where the random walks are transient but with a sublinear speed of the order $n^\kappa$ for some $\kappa \in (0,1)$ and prove a quenched hydrodynamic limit for the system of random walks with time scaled by $n^{1/\kappa}$ and space scaled by $n$. The most interesting feature of the hydrodynamic limit is that the influence of the environment does not average out under the hydrodynamic scaling; that is, the asymptotic particle density depends on the specific environment chosen. The hydrodynamic limit for the system of RWRE is obtained by first proving a hydrodynamic limit for a system of independent particles in a directed trap environment.