Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Large deviations for voter model occupation times in two dimensions

G. Maillard and T. Mountford

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Abstract

We study the decay rate of large deviation probabilities of occupation times, up to time t, for the voter model η: ℤ2×[0, ∞)→{0, 1} with simple random walk transition kernel, starting from a Bernoulli product distribution with density ρ∈(0, 1). In [Probab. Theory Related Fields 77 (1988) 401–413], Bramson, Cox and Griffeath showed that the decay rate order lies in [log(t), log2(t)].

In this paper, we establish the true decay rates depending on the level. We show that the decay rates are log2(t) when the deviation from ρ is maximal (i.e., η≡0 or 1), and log(t) in all other situations. This answers some conjectures in [Probab. Theory Related Fields 77 (1988) 401–413] and confirms nonrigorous analysis carried out in [Phys. Rev. E 53 (1996) 3078–3087], [J. Phys. A 31 (1998) 5413–5429] and [J. Phys. A 31 (1998) L209–L215].

Résumé

On étudie le taux de décroissance des probabilités de grandes déviations des temps d’occupation, jusqu’à l’instant t, du modèle du votant η: ℤ2×[0, ∞)→{0, 1} ayant le noyau de transition d’une marche aléatoire simple et partant d’une distribution produit de Bernoulli de paramètre ρ∈(0, 1). Dans [Probab. Theory Related Fields 77 (1988) 401–413], Bramson, Cox et Griffeath ont montré que l’ordre du taux de décroissance se situe dans [log(t), log2(t)].

Dans cet article, nous établissons les taux de décroissance exacts dépendant du niveau. On prouve que les taux de décroissance sont log2(t) lorsque la déviation de ρ est maximale (i.e., η≡0 ou 1), et log(t) dans toutes les autres situations. Ceci répond à une conjecture de [Probab. Theory Related Fields 77 (1988) 401–413] et confirme l’analyse non rigoureuse effectuée dans [Phys. Rev. E 53 (1996) 3078–3087], [J. Phys. A 31 (1998) 5413–5429] et [J. Phys. A 31 (1998) L209–L215].

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 2 (2009), 577-588.

Dates
First available in Project Euclid: 29 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1241024681

Digital Object Identifier
doi:10.1214/08-AIHP178

Mathematical Reviews number (MathSciNet)
MR2521414

Zentralblatt MATH identifier
1173.60342

Subjects
Primary: 60F10: Large deviations 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J25: Continuous-time Markov processes on general state spaces

Keywords
Voter model Large deviations

Citation

Maillard, G.; Mountford, T. Large deviations for voter model occupation times in two dimensions. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 2, 577--588. doi:10.1214/08-AIHP178. https://projecteuclid.org/euclid.aihp/1241024681


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References

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