Non-Fixation for Activated Random Walks with Small Particle Density

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).

Biography
Riddhipratim Basu obtained his Ph.D. from University of California, Berkeley in 2015 under supervision of Allan Sly and is currently a Szegö Assistant Professor of Mathematics at Stanford University. His research interest is in discrete probability with special focus on random spatial processes.