Activated Random Walk on a cycle

TL;DR

This paper analyzes the Activated Random Walk model on a cycle, showing how the fixation time scales linearly or exponentially with system size depending on particle density and sleep rate, reflecting phase transitions known from infinite systems.

Contribution

It establishes the asymptotic fixation time behavior of ARW on finite cycles, connecting finite system dynamics to known phase transitions in infinite systems.

Findings

Fixation time is linear in system size for low density and high sleep rate.

Fixation time is exponential in system size for high density and low sleep rate.

Results reflect the phase transition between fixation and non-fixation in infinite ARW systems.

Abstract

We consider Activated Random Walk (ARW), a particle system with mass conservation, on the cycle $\mathbb{Z}/n\mathbb{Z}$. One starts with a mass density $\mu>0$ of initially active particles, each of which performs a simple symmetric random walk at rate one and falls asleep at rate $\lambda>0.$ 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 $\mu$ and $\lambda$. On the finite graph $\mathbb{Z}/n\mathbb{Z}$, unless there are more than $n$ particles, the process fixates (reaches an absorbing state) almost surely in finite time. 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 by Rolla, Sidoravicius (2012).