Activated random walk on a cycle

Abstract

We consider Activated Random Walk (ARW), a particle system with mass conservation, on the cycle Z/nZ. One starts with a mass density μ>0 of initially active particles, each of which performs a simple 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. There have been several recent results concerning fixation/non-fixation of the ARW dynamics on infinite systems depending on the parameters μ and λ. On the finite graph Z/nZ, unless there are more than n particles, the process fixates (reaches an absorbing state) almost surely in finite time. In a first rigorous result for a finite system, establishing well known beliefs in the statistical physics literature, we show that the number of steps the process takes to fixate is linear in n (up to poly-logarithmic terms), when the density is sufficiently low compared to the sleep rate, and exponential in n when the sleep rate is sufficiently small compared to the density, reflecting the fixation/non-fixation phase transition in the corresponding infinite system as established in (Invent. Math. 188 (2012) 127–150).