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

摘要

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 (Electron. J. Probab. 15 (2010) 1024-1040). 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 is an element of (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.