How far do Activated Random Walkers spread from a single source?

Abstract

Unlike many particle systems, Activated Random Walk has nontrivial behavior even in one spatial dimension. We prove inner and outer bounds on the spread of n activated random walkers from a single source in Z. The inner bound involves a comparison with the stationary distribution of activated random walkers on a finite interval, while the outer bound involves a comparison with the stabilization of an infinite Bernoulli configuration of activated random walkers on Z.

Exact sampling and fast mixing of Activated Random Walk

“…Our first order of business is to check that doing so does not change the distribution of (σ t ) t∈N . For this purpose we will use a lemma from [22].…”

“…For instance, we expect that the ARW stationary distribution µ V on a finite subset V ⊂ Z d has an infinite-volume limit µ, and that its mean equals ζ c . We also expect the microstructure of finite ARW clusters (such as the cluster of sleeping particles formed by stabilizing n chips at the origin in Z d , studied in [22]) to converge to µ as n → ∞. These conjectures are detailed in the forthcoming paper [23].…”