Probability Seminar
Activated Random Walk (ARW) is an interacting particle system that approximates the Stochastic Sandpile Model (SSM), a paradigm example of a model of self-organized criticality. On $\mathbb{Z}$, one starts with a mass density $\mu$ of initially active particles each of which performs a continuous time nearest neighbour symmetric random walk at rate one and falls asleep at rate $\lambda>0$. Sleepy particles become active on coming in contact with active particles. I shall describe a recent joint work with Shirshendu Ganguly and Christopher Hoffman where we use a novel renormalized variant of the Diaconis-Fulton construction of the process and the associated Abelian property to show that even at arbitrarily small positive desnity of particles, the system does not fixate provided the sleep rate $\lambda$ is sufficiently small. This answers positively two open questions from Rolla and Sidoravicius (Invent. Math., 2012).