Probability Theory and Related Fields

, Volume 77, Issue 3, pp 401–413

Occupation time large deviations of the voter model

  • Maury Bramson
  • J. Theodore Cox
  • David Griffeath
Article

Abstract

This paper is a sequel to [5] and [6]. We continue our study of occupation time large deviation probabilities for some simple infinite particle systems by analysing the so-called voter model ζt (see e.g., [11] or [8]). In keeping with our previous results, we show that the large deviations are “classical” in high dimensions (d≧5 for ζt) but “fat” in low dimensions (d≦4). Interaction distinguishes the voter model from the independent particle systems of [5] and [6], and consequently exact computations no longer seem feasible. Instead, we derive upper and lower bounds which capture the asymptotic decay rate of the large deviation tails.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arratia R.: Symmetric exclusion processes: a comparison inequality and a large deviation result. Ann. Probab. 13, 53–61 (1985)Google Scholar
  2. 2.
    Bramson M., Cox J. T., Griffeath D.: Consolidation rates for two interacting systems in the plane. Probab. Th. Rel. Fields 73, 613–625 (1986)Google Scholar
  3. 3.
    Bramson M., Griffeath D: Asymptotics for interacting particle systems on Zd. Z. Wahrscheinlichkeitstheor. Verw. Geb. 53, 183–196 (1980)Google Scholar
  4. 4.
    Cox J.T., Griffeath D.: Occupation time limit theorems for the voter model. Ann. Probab. 11, 876–893 (1983)Google Scholar
  5. 5.
    Cox J.T., Griffeath D.: Large deviations for Poisson systems of independent random walks. Z. Wahrscheinlichkeitstheor. Verw. Geb. 66, 543–558 (1984)Google Scholar
  6. 6.
    Cox J.T., Griffeath, D.: Occupation times for critical branching Brownian motions. Ann. Probab. 13, 1108–1132 (1985)Google Scholar
  7. 7.
    Cox J.T., Griffeath D.: Large deviations for some infinite particle system occupation times. Contemp. Math. 41, 43–54 (1985)Google Scholar
  8. 8.
    Cox J.T., Griffeath D.: Diffusive clustering in the two dimensional voter model. Ann. Probab. 14, 347–370 (1986)Google Scholar
  9. 9.
    Donsker M., Varadhan S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time, IV. Comm. Pure Appl. Math. 36, 183–312 (1983)Google Scholar
  10. 10.
    Van den Berg, J., Kesten H.: Inequalities with applications to percolation and reliability. J. Appl. Probab. 22, 556–599 (1985)Google Scholar
  11. 11.
    Liggett T.M.: Interacting Particle Systems. Berlin Heidelberg New York: Springer 1985Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Maury Bramson
    • 1
  • J. Theodore Cox
    • 2
  • David Griffeath
    • 3
  1. 1.Department of MathematicsUniversity of MinnesotaMineapolisUSA
  2. 2.Department of MathematicsSyracuse UniversitySyracuseUSA
  3. 3.Department of MathematicsUniversity of WisconsinMadisonUSA

Personalised recommendations