Random walk in a high density dynamic random environment

Abstract

The goal of this note is to prove a law of large numbers for the empirical speed of a green particle that performs a random walk on top of a field of red particles which themselves perform independent simple random walks on ${\mathbb{Z}}^d$, $d\ge 1$. The red particles jump at rate 1 and are in a Poisson equilibrium with density $\mu$. The green particle also jumps at rate 1, but uses different transition kernels $p^{\prime }$ and $p^{\prime \prime }$ depending on whether it sees a red particle or not. It is shown that, in the limit as $\mu \to \infty$, the speed of the green particle tends to the average jump under $p^{\prime }$. This result is far from surprising, but it is non-trivial to prove. The proof that is given in this note is based on techniques that were developed in \cite{KeSi} to deal with spread-of-infection models. The main difficulty is that, due to particle conservation, space-time correlations in the field of red particles decay slowly. This places the problem in a class of random walks in dynamic random environments for which scaling laws are hard to obtain.

14 pages, 1 figure