Law of large numbers for random walks on attractive spin-flip dynamics

TL;DR

This paper establishes a law of large numbers for random walks in attractive dynamic environments, including the contact process, and provides conditions for relaxing initial state assumptions and analyzing large deviations.

Contribution

It introduces a law of large numbers for random walks on attractive spin-flip dynamics and extends results to the contact process with new speed properties.

Findings

Law of large numbers proven for random walks on attractive environments.

Sufficient mixing conditions allow relaxation of initial state assumptions.

Derived estimates on large deviation behavior and properties of the walk's speed.

Abstract

We prove a law of large numbers for certain random walks on certain attractive dynamic random environments when initialised from all sites equal to the same state. This result applies to random walks on $\mathbb{Z}^d$ with $d\geq1$. We further provide sufficient mixing conditions under which the assumption on the initial state can be relaxed, and obtain estimates on the large deviation behaviour of the random walk. As prime example we study the random walk on the contact process, for which we obtain a law of large numbers in arbitrary dimension. For this model, further properties about the speed are derived.