Large deviations for voter model occupation times in two dimensions
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 2, p. 577-588

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].

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].

DOI : https://doi.org/10.1214/08-AIHP178
Classification:  60F10,  60K35,  60J25
Keywords: voter model, large deviations
@article{AIHPB_2009__45_2_577_0,
     author = {Maillard, G. and Mountford, T.},
     title = {Large deviations for voter model occupation times in two dimensions},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {2},
     year = {2009},
     pages = {577-588},
     doi = {10.1214/08-AIHP178},
     zbl = {1173.60342},
     mrnumber = {2521414},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_2_577_0}
}
Maillard, G.; Mountford, T. Large deviations for voter model occupation times in two dimensions. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 2, pp. 577-588. doi : 10.1214/08-AIHP178. http://www.numdam.org/item/AIHPB_2009__45_2_577_0/

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