Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Non-fixation for biased Activated Random Walks

L. T. Rolla and L. Tournier

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We prove that the model of Activated Random Walks on $\mathbb{Z}^{d}$ with biased jump distribution does not fixate for any positive density, if the sleep rate is small enough, as well as for any finite sleep rate, if the density is close enough to $1$. The proof uses a new criterion for non-fixation. We provide a pathwise construction of the process, of independent interest, used in the proof of this non-fixation criterion.


On démontre que le modèle de marches aléatoires activées sur $\mathbb{Z}^{d}$ avec distribution de saut biaisée ne se fixe pas, quelle que soit la densité initiale si le taux de désactivation est suffisamment bas, ou quel que soit le taux (fini) de désactivation si la densité initiale est suffisamment proche de $1$. La démonstration fait appel à un nouveau critère de non-fixation. On fournit également une construction d’une version trajectorielle du processus, qui est utilisée dans la preuve de ce critère et qui présente un intérêt indépendant.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 2 (2018), 938-951.

Received: 20 October 2015
Revised: 24 November 2016
Accepted: 7 March 2017
First available in Project Euclid: 25 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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] 82C26: Dynamic and nonequilibrium phase transitions (general)

Interacting particle systems Activated Random Walks Absorbing-state Phase transition


Rolla, L. T.; Tournier, L. Non-fixation for biased Activated Random Walks. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 2, 938--951. doi:10.1214/17-AIHP827.

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