Active Phase for Activated Random Walks on the Lattice in all Dimensions

Abstract

We show that the critical density of the Activated Random Walk model on ${\mathbb{Z}}^d$ is strictly less than one when the sleep rate $\lambda$ is small enough, and tends to $0$ when $\lambda \to 0$, in any dimension $d⩾1$. As far as we know, the result is new for $d=2$. We prove this by showing that, for high enough density and small enough sleep rate, the stabilization time of the model on the $d$-dimensional torus is exponentially large. To do so, we fix the the set of sites where the particles eventually fall asleep, which reduces the problem to a simpler model with density one. Taking advantage of the Abelian property of the model, we show that the stabilization time stochastically dominates the escape time of a one-dimensional random walk with a negative drift. We then check that this slow phase for the finite volume dynamics implies the existence of an active phase on the infinite lattice.

arXiv comment27 pages, new version with minor corrections

References in corpus (12)

• The spread of a rumor or infection in a moving population
• Signatures of Self-Organised Criticality in an Ultracold Atomic Gas
• Phase transition for the frog model
• Activated Random Walkers: Facts, Conjectures and Challenges
• Absorbing-State Phase Transition for Driven-Dissipative Stochastic Dynamics on $Z$
• Recurrence and transience for the frog model on trees
• Non-equilibrium Phase Transitions: Activated Random Walks at Criticality
• Stochastic orders and the frog model
• Nonfixation for Activated Random Walks
• Activated Random Walks on ${\mathbb{Z}}^d$
• Non-fixation for Biased Activated Random Walks
• Generalized Hammersley Process and Phase Transition for Activated Random Walk Models