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 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 the critical mass density for fixation is strictly less than one. Moreover, the critical density goes to zero as tends to zero. This settles a long standing open question.