Absorbing-state phase transition and activated random walks with unbounded capacities

TL;DR

This paper investigates the phase transition behavior of a generalized activated random walk model with unbounded site capacities on vertex transitive graphs, establishing conditions for fixation or ongoing activity based on the distribution of capacities.

Contribution

It extends the classical activated random walk model by incorporating random, unbounded site capacities and analyzes the conditions for phase transition and fixation.

Findings

Existence of an absorbing-state phase transition when expected capacity is finite.

Model fixates for all positive particle densities when expected capacity is infinite.

Provides bounds for the critical density matching classical ARW results.

Abstract

In this article, we study the existence of an absorbing-state phase transition of an Abelian process that generalises the Activated Random Walk (ARW). Given a vertex transitive $G=(V,E)$, we associate to each site $x \in V$ a capacity $w_x \ge 0$, which describes how many inactive particles $x$ can hold, where $\{w_x\}_{x \in V}$ is a collection of i.i.d random variables. When $G$ is an amenable graph, we prove that if $\mathbb E[w_x]<\infty$, the model goes through an absorbing state phase transition and if $\mathbb E[w_x]=\infty$, the model fixates for all $\lambda>0$. Moreover, in the former case, we provide bounds for the critical density that match the ones available in the classical Activated Random Walk.