Non-fixation for conservative stochastic dynamics on the line

TL;DR

This paper studies the Activated Random Walk model on the line, showing that for small sleep rates, the critical initial density for the system to fixate is less than one and approaches zero as sleep rate decreases, resolving a long-standing open problem.

Contribution

It proves that the critical density for fixation is less than one for small sleep rates and tends to zero as the sleep rate approaches zero, settling a long-standing open question.

Findings

Critical density for fixation is less than one for small λ.

Critical density approaches zero as λ tends to zero.

The results settle a long-standing open problem.

Abstract

We consider Activated Random Walk (ARW), a model which generalizes the Stochastic Sandpile, one of the canonical examples of self organized criticality. Informally ARW is a particle system on $\mathbb{Z}$ with mass conservation. One starts with a mass density $\mu>0$ of initially active particles, each of which performs a symmetric random walk at rate one and falls asleep at rate $\lambda>0$. Sleepy particles become active on coming in contact with other active particles. We investigate the question of fixation/non-fixation of the process, and show for small enough $\lambda$, the critical mass density for fixation is strictly less than one. Moreover, the critical density goes to zero as $\lambda$ tends to zero. This settles a long standing open question.