Explosivity in 1-d Activated Random Walk

TL;DR

This paper proves that in one-dimensional Activated Random Walks, a small initial activation can cause an infinite cascade of activity in supercritical states, confirming a key conjecture and highlighting the universality of phase transitions.

Contribution

It extends previous results to ergodic initial distributions, establishing explosivity in 1D Activated Random Walks and confirming a conjecture of Rolla.

Findings

Activation triggers infinite avalanches in supercritical states

System remains active almost surely with positive initial activity

Supports universality of phase transition in 1D Activated Random Walks

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 $\mathbb{Z}$.