Active Phase for Activated Random Walk on Z

TL;DR

This work proves that Activated Random Walk on the one-dimensional lattice $\mathbb{Z}$ possesses a nontrivial active phase for any finite sleep rate $\lambda$ when the initial density $\zeta$ is chosen sufficiently close to 1. The authors develop a block-structured carpet-hole toppling procedure to circumvent long-range correlations, coupled with a mass-balance framework and a careful conditioning via filtrations to obtain a robust single-block estimate. The key technical achievement is an exponential bound on the number of frozen free particles within large finite intervals, which, together with a stabilization criterion, implies a.s. perpetual activity for high-density initial configurations. This advances the understanding of universality and self-organized criticality in Abelian networks and extends prior results to large sleep rates in the 1D ARW model. The methods—site-wise representations, block arguments, and intricate probabilistic couplings—provide a template for similar analyses in related interacting particle systems.

Abstract

We consider the Activated Random Walk model on $\mathbb{Z}$. In this model, each particle performs a continuous-time simple symmetric random walk, and falls asleep at rate $λ$. A sleeping particle does not move but it is reactivated in the presence of another particle. We show that for any sleep rate $λ< \infty$ if the density $ ζ$ is close enough to $1$ then the system stays active.

Figures (1)

• Figure 1: The phase diagram for ARW on $\mathop{\mathrm{\mathbb{Z}}}\nolimits$. In this paper we show that for every value of the sleep rate $\lambda$ the critical density $\zeta$ is less than one.

Theorems & Definitions (29)

• Theorem 1.1
• Lemma 2.4
• proof
• Theorem 2.5
• Lemma 3.3
• proof
• Proposition 3.5
• proof : Proof of Theorem \ref{['main_thm']}
• Lemma 4.3
• proof