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Activated random walk on a cycle. (English. French summary) Zbl 1442.60097

Summary: 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. 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 [L. T. Rolla and V. Sidoravicius, Invent. Math. 188, No. 1, 127–150 (2012; Zbl 1242.60104)].

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics

Citations:

Zbl 1242.60104
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References:

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