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

TL;DR

This paper investigates the spread of activated random walkers in one dimension, providing bounds on their propagation from a single source using comparisons with stationary and stabilized configurations.

Contribution

It introduces new bounds on the spread of activated random walkers in one dimension, utilizing novel comparison techniques with stationary and stabilized configurations.

Findings

Inner bounds based on stationary distributions

Outer bounds based on stabilization of Bernoulli configurations

Provides a framework for understanding particle spread in 1D systems

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.