Translator Disclaimer
August 2019 Active phase for activated random walks on $\mathbb{Z}^{d}$, $d\geq3$, with density less than one and arbitrary sleeping rate
Lorenzo Taggi
Ann. Inst. H. Poincaré Probab. Statist. 55(3): 1751-1764 (August 2019). DOI: 10.1214/18-AIHP933

Abstract

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.

Citation

Download Citation

Lorenzo Taggi. "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 (3) 1751 - 1764, August 2019. https://doi.org/10.1214/18-AIHP933

Information

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

zbMATH: 07133736
MathSciNet: MR4010950
Digital Object Identifier: 10.1214/18-AIHP933

Subjects:
Primary: 82C22
Secondary: 60K35, 82C26

Rights: Copyright © 2019 Institut Henri Poincaré

JOURNAL ARTICLE
14 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.55 • No. 3 • August 2019
Back to Top