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].
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].
Mots clés : 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}, pages = {577--588}, publisher = {Gauthier-Villars}, volume = {45}, number = {2}, year = {2009}, doi = {10.1214/08-AIHP178}, mrnumber = {2521414}, zbl = {1173.60342}, language = {en}, url = {http://www.numdam.org/articles/10.1214/08-AIHP178/} }
TY - JOUR AU - Maillard, G. AU - Mountford, T. TI - Large deviations for voter model occupation times in two dimensions JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2009 SP - 577 EP - 588 VL - 45 IS - 2 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/08-AIHP178/ DO - 10.1214/08-AIHP178 LA - en ID - AIHPB_2009__45_2_577_0 ER -
%0 Journal Article %A Maillard, G. %A Mountford, T. %T Large deviations for voter model occupation times in two dimensions %J Annales de l'I.H.P. Probabilités et statistiques %D 2009 %P 577-588 %V 45 %N 2 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/08-AIHP178/ %R 10.1214/08-AIHP178 %G en %F 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, Tome 45 (2009) no. 2, pp. 577-588. doi : 10.1214/08-AIHP178. http://www.numdam.org/articles/10.1214/08-AIHP178/
[1] Coarsening and persistence in the voter model. Phys. Rev. E 53 (1996) 3078-3087.
, and .[2] Occupation time large deviations of the voter model. Probab. Theory Related Fields 77 (1988) 401-413. | MR | Zbl
, and .[3] A model for spatial conflict. Biometrika 60 (1973) 581-588. | MR | Zbl
and .[4] Some limit theorems for voter model occupation times. Ann. Probab. 16 (1988) 1559-1569. | MR | Zbl
.[5] Occupation time limit theorems for the voter model. Ann. Probab. 11 (1983) 876-893. | MR | Zbl
and .[6] Diffusive clustering in the two dimensional voter model. Ann. Probab. 14 (1986) 347-370. | MR | Zbl
and .[7] Large deviations and nontrivial exponents in coarsening systems. J. Phys. A 31 (1998) 5413-5429. | MR | Zbl
and .[8] Lecture Notes on Particle Systems and Percolation. Belmont, Wadsworth, CA, 1988. | MR | Zbl
.[9] Probability: Theory and Examples, 3rd edition. Duxbury Press, Belmont, CA, 2005. | MR | Zbl
.[10] Large Deviations. Fields Institute Monographs 14. Amer. Math. Soc., Providence, RI, 2000. | MR | Zbl
.[11] Ergodic theorems for weakly interacting infinite systems and the voter model. Ann. Probab. 3 (1975) 643-663. | MR | Zbl
and .[12] Persistence in the voter model: Continuum reaction-diffusion approach. J. Phys. A 31 (1998) L209-L215. | MR | Zbl
and .[13] Intersections of Random Walks. Birkhäuser, Boston, 1991. | MR | Zbl
.[14] Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften 276. Springer, New York, 1985. | MR | Zbl
.Cité par Sources :