Large deviations for voter model occupation times in two dimensions

G. Maillard; T. Mountford

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 2, page 577-588
  • ISSN: 0246-0203

Abstract

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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 Fields77 (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 Fields77 (1988) 401–413] and confirms nonrigorous analysis carried out in [Phys. Rev. E53 (1996) 3078–3087], [J. Phys. A31 (1998) 5413–5429] and [J. Phys. A31 (1998) L209–L215].

How to cite

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Maillard, G., and Mountford, T.. "Large deviations for voter model occupation times in two dimensions." Annales de l'I.H.P. Probabilités et statistiques 45.2 (2009): 577-588. <http://eudml.org/doc/78034>.

@article{Maillard2009,
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 Fields77 (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 Fields77 (1988) 401–413] and confirms nonrigorous analysis carried out in [Phys. Rev. E53 (1996) 3078–3087], [J. Phys. A31 (1998) 5413–5429] and [J. Phys. A31 (1998) L209–L215].},
author = {Maillard, G., Mountford, T.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {voter model; large deviations; Voter model},
language = {eng},
number = {2},
pages = {577-588},
publisher = {Gauthier-Villars},
title = {Large deviations for voter model occupation times in two dimensions},
url = {http://eudml.org/doc/78034},
volume = {45},
year = {2009},
}

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
PB - Gauthier-Villars
VL - 45
IS - 2
SP - 577
EP - 588
AB - 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 Fields77 (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 Fields77 (1988) 401–413] and confirms nonrigorous analysis carried out in [Phys. Rev. E53 (1996) 3078–3087], [J. Phys. A31 (1998) 5413–5429] and [J. Phys. A31 (1998) L209–L215].
LA - eng
KW - voter model; large deviations; Voter model
UR - http://eudml.org/doc/78034
ER -

References

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  1. [1] E. Ben-Naim, L. Frachebourg and P. L. Krapivsky. Coarsening and persistence in the voter model. Phys. Rev. E 53 (1996) 3078–3087. 
  2. [2] M. Bramson, J. T. Cox and D. Griffeath. Occupation time large deviations of the voter model. Probab. Theory Related Fields 77 (1988) 401–413. Zbl0621.60107MR931506
  3. [3] P. Clifford and A. Sudbury. A model for spatial conflict. Biometrika 60 (1973) 581–588. Zbl0272.60072MR343950
  4. [4] J. T. Cox. Some limit theorems for voter model occupation times. Ann. Probab. 16 (1988) 1559–1569. Zbl0656.60105MR958202
  5. [5] J. T. Cox and D. Griffeath. Occupation time limit theorems for the voter model. Ann. Probab. 11 (1983) 876–893. Zbl0527.60095MR714952
  6. [6] J. T. Cox and D. Griffeath. Diffusive clustering in the two dimensional voter model. Ann. Probab. 14 (1986) 347–370. Zbl0658.60131MR832014
  7. [7] I. Dornic and C. Godrèche. Large deviations and nontrivial exponents in coarsening systems. J. Phys. A 31 (1998) 5413–5429. Zbl0954.82010MR1632861
  8. [8] R. Durrett. Lecture Notes on Particle Systems and Percolation. Belmont, Wadsworth, CA, 1988. Zbl0659.60129MR940469
  9. [9] R. Durrett. Probability: Theory and Examples, 3rd edition. Duxbury Press, Belmont, CA, 2005. Zbl0709.60002MR1609153
  10. [10] F. den Hollander. Large Deviations. Fields Institute Monographs 14. Amer. Math. Soc., Providence, RI, 2000. Zbl0949.60001MR1739680
  11. [11] R. A. Holley and T. M. Liggett. Ergodic theorems for weakly interacting infinite systems and the voter model. Ann. Probab. 3 (1975) 643–663. Zbl0367.60115MR402985
  12. [12] M. Howard and C. Godrèche. Persistence in the voter model: Continuum reaction-diffusion approach. J. Phys. A 31 (1998) L209–L215. Zbl0925.60126MR1628504
  13. [13] G. F. Lawler. Intersections of Random Walks. Birkhäuser, Boston, 1991. Zbl0925.60078MR1117680
  14. [14] T. M. Liggett. Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften 276. Springer, New York, 1985. Zbl0559.60078MR776231

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