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

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∈V a capacity wx≥0, which describes how many inactive particles x can hold, where {wx}x∈V is a collection of i.i.d random variables. When G is an amenable graph, we prove that if E[wx]<∞, the model goes through an absorbing state phase transition and if E[wx]=∞, the model fixates for all λ>0. Moreover, in the former case, we provide bounds for the critical density that match the ones available in the classical Activated Random Walk.