Non-fixation for conservative stochastic dynamics on the line

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 $μ>0$ of initially active particles, each of which performs a symmetric random walk at rate one and falls asleep at rate $λ>0$. 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 $λ$, the critical mass density for fixation is strictly less than one. Moreover, the critical density goes to zero as $λ$ tends to zero. This settles a long standing open question.

29 pages, 4 figures. Final version. To appear in Comm. Math. Phys