Systems of one-dimensional random walks in a common random environment

TL;DR

This paper studies a system of independent one-dimensional random walks in a shared random environment, establishing probabilistic bounds, laws of large numbers, and identifying stationary distributions for the system.

Contribution

It provides new upper bounds on quenched probabilities, proves a uniform law of large numbers, and characterizes stationary distributions for the system.

Findings

Upper bounds on quenched probabilities for walks lagging behind expected position

Proof of a uniform law of large numbers for the system

Identification of stationary distributions parameterized by particle density

Abstract

We consider a system of independent one-dimensional random walks in a common random environment under the condition that the random walks are transient with positive speed $v_P$. We give upper bounds on the quenched probability that at least one of the random walks started in the interval $[An, Bn]$ has traveled a distance of less than $(v_P - \epsilon)n$. This leads to both a uniform law of large numbers and a hydrodynamic limit. We also identify a family of distributions on the configuration of particles (parameterized by particle density) which are stationary under the (quenched) dynamics of the random walks and show that these are the limiting distributions for the system when started from a certain natural collection of distributions.