Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Activated random walk on a cycle

Riddhipratim Basu, Shirshendu Ganguly, Christopher Hoffman, and Jacob Richey

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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 (Invent. Math. 188 (2012) 127–150).


Nous considérons la marche aléatoire activée (Activated Random Walk, ARW), un système de particules avec conservation de masse sur le cycle $\mathbb{Z}/n\mathbb{Z}$. Partant d’un état initial avec une densité $\mu >0$ de particules actives, chacune d’entre elles évolue selon une marche simple symétrique à taux $1$, et s’endort à taux $\lambda >0$. Les particules endormies deviennent actives lorsqu’elles entrent en contact avec d’autres particules actives. Plusieurs résultats récents se sont penchés sur la fixation ou la non-fixation de la dynamique en volume infini, en fonction des paramètres $\mu $ et $\lambda $. Sur le graphe fini $\mathbb{Z}/n\mathbb{Z}$, à moins qu’il y ait plus de $n$ particules, le processus se fixe (en atteignant un état absorbant) presque sûrement en temps fini. Nous établissons un premier résultat rigoureux sur ces systèmes finis, confirmant des prédictions bien connues de la littérature de physique statistique, en montrant que le nombre d’étapes avant fixation est linéaire en $n$ (à des termes poly-logarithmiques près) lorsque la densité est suffisamment petite par rapport au taux d’endormissement, et exponentielle en $n$ lorsque le taux d’endormissement est suffisamment petit par rapport à la densité, ce qui reflète la transition de phase entre fixation et non-fixation établie dans (Invent. Math. 188 (2012) 127–150) pour le système infini.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1258-1277.

Received: 21 February 2018
Revised: 17 May 2018
Accepted: 24 June 2018
First available in Project Euclid: 25 September 2019

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Mathematical Reviews number (MathSciNet)

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs 82C22: Interacting particle systems [See also 60K35]

Activated random walk Diaconis–Fulton representation Abelian property Self-ogranized criticality


Basu, Riddhipratim; Ganguly, Shirshendu; Hoffman, Christopher; Richey, Jacob. Activated random walk on a cycle. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1258--1277. doi:10.1214/18-AIHP918.

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