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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Abstract

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.

Résumé

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1524643235

Digital Object Identifier
doi:10.1214/17-AIHP827

Mathematical Reviews number (MathSciNet)
MR3795072

Zentralblatt MATH identifier
06897974

Subjects
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)

Keywords
Interacting particle systems Activated Random Walks Absorbing-state Phase transition

Citation

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. https://projecteuclid.org/euclid.aihp/1524643235


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References

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