Random walk in a high density dynamic random environment

TL;DR

This paper proves a law of large numbers for a green particle's speed in a dynamic environment of red particles performing independent random walks, showing it converges to the average jump as red particle density increases.

Contribution

It establishes a non-trivial law of large numbers for a random walk in a high-density dynamic environment, extending techniques from infection spread models.

Findings

Green particle's speed converges to average jump under certain conditions.

The proof handles slow decay of correlations in the environment.

Results are relevant for understanding random walks in complex dynamic systems.

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 $\Z^d$, $d \geq 1$. 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 $p'$ and $p''$ 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 $p'$. 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.