Universality of Activated Random Walk

Abstract

Activated random walk is an interacting particle system proposed as a tractable model of self-organized criticality. Particles are active or sleeping: active particles move by random walk and may fall asleep when isolated, while sleeping particles are reactivated by incoming active particles. A central theme is universality: the critical density and critical local state should be robust under changes in initial condition, driving mechanism, and boundary condition. This dissertation develops two one-dimensional results under the theme of universality. First, for a supercritical Bernoulli field of sleeping particles on $\mathbb Z$ with only one initially active particle, the probability of fixation is strictly between zero and one. Thus the supercritical active phase is not merely an artifact of starting all particles active. Second, for the point-source model started from $n$ active particles at the origin, the probability that a bulk site contains a sleeping particle after stabilization converges to the common critical density. Consequently, any subsequential microscopic local limit of the point-source laws is shift invariant and has the correct one-site density.