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 $${\mu > 0}$$ 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.