Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Active phase for activated random walks on $\mathbb{Z}^{d}$, $d\geq3$, with density less than one and arbitrary sleeping rate

Lorenzo Taggi

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It has been conjectured that the critical density of the Activated Random Walk model is strictly less than one for any value of the sleeping rate. We prove this conjecture on $\mathbb{Z}^{d}$ when $d\geq3$ and, more generally, on graphs where the random walk is transient. Moreover, we establish the occurrence of a phase transition on non-amenable graphs, extending previous results which require that the graph is amenable or a regular tree.


Il a été conjecturé que la densité critique pour le modèle de marches aléatoires activées était strictement inférieur à 1 pour toute valeur du taux d’endormissement. Nous démontrons cette conjecture pour $\mathbb{Z}^{d}$ quand $d\geq3$ et, plus généralement, pour les graphes sur lesquels la marche aléatoire est transitoire. De plus, nous montrons l’existence d’une transition de phase pour les graphes non moyennables, généralisant ainsi des résultats antérieurs qui demandaient que le graphe soit moyennable ou un arbre régulier.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1751-1764.

Received: 3 January 2018
Revised: 5 June 2018
Accepted: 17 September 2018
First available in Project Euclid: 25 September 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82C22: Interacting particle systems [See also 60K35]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C26: Dynamic and nonequilibrium phase transitions (general)

Interacting particle systems Abelian networks Absorbing-state phase transition Self-organized criticality


Taggi, Lorenzo. Active phase for activated random walks on $\mathbb{Z}^{d}$, $d\geq3$, with density less than one and arbitrary sleeping rate. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1751--1764. doi:10.1214/18-AIHP933.

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