Explosivity in 1-d Activated Random Walk

TL;DR

The paper addresses the explosivity of Activated Random Walk on $\mathbb{Z}$ under ergodic initial data and a universal density threshold $\rho_c(\lambda,d)$. It develops a nucleation-avalanching argument based on a site-wise stack construction, the preemptive abelian property, and cutoff/large-deviation control using the quantities $X(n,\sigma)$, $Y(n,\sigma)$ and the stabilizing odometer. The main contributions establish explosivity for $\rho>\rho_c(\lambda,d)$ and non-explosivity for $\rho<\rho_c(\lambda,d)$, thereby completing Rolla's conjecture in dimension $d=1$ and extending results from i.i.d. to ergodic initial distributions. This work reinforces the universality of the ARW phase transition and links fixed-energy behavior with driven-dissipative/continuum perspectives.

Abstract

We show that Activated Random Walk on $\mathbb{Z}$ is explosive above criticality. That is, activating a single particle in a supercritical state of sleeping particles triggers an infinite avalanche of activity with positive probability. This extends the same result recently proven by Brown, Hoffman, and Son for i.i.d. initial distributions to the setting of ergodic ones, thus completing the proof of a conjecture of Rolla's in dimension one. As a corollary we obtain that, for supercritical ergodic initial distributions with any positive density of particles initially active, the system will stay active almost surely. Our result is another piece of evidence attesting to the universality of the phase transition of Activated Random Walk on $\mathbb{Z}$.

Theorems & Definitions (12)

• Theorem 1
• Remark 2
• Lemma 3
• proof
• Definition 4: $X(n, \sigma)$, $Y(n, \sigma)$, $X(n, \sigma,u_0)$, $Y(n,\sigma,u_0)$
• Proposition 5
• proof
• Lemma 6
• proof
• Proposition 7