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).
Categories: math.PR cond-mat.stat-mech math-ph math.CO math.MP