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Occupation time large deviations of the voter model
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  • Published: March 1988

Occupation time large deviations of the voter model

  • Maury Bramson1,
  • J. Theodore Cox2 &
  • David Griffeath3 

Probability Theory and Related Fields volume 77, pages 401–413 (1988)Cite this article

  • 138 Accesses

  • 15 Citations

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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.

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References

  1. Arratia R.: Symmetric exclusion processes: a comparison inequality and a large deviation result. Ann. Probab. 13, 53–61 (1985)

    Google Scholar 

  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. Bramson M., Griffeath D: Asymptotics for interacting particle systems on Zd. Z. Wahrscheinlichkeitstheor. Verw. Geb. 53, 183–196 (1980)

    Google Scholar 

  4. Cox J.T., Griffeath D.: Occupation time limit theorems for the voter model. Ann. Probab. 11, 876–893 (1983)

    Google Scholar 

  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. Cox J.T., Griffeath, D.: Occupation times for critical branching Brownian motions. Ann. Probab. 13, 1108–1132 (1985)

    Google Scholar 

  7. Cox J.T., Griffeath D.: Large deviations for some infinite particle system occupation times. Contemp. Math. 41, 43–54 (1985)

    Google Scholar 

  8. Cox J.T., Griffeath D.: Diffusive clustering in the two dimensional voter model. Ann. Probab. 14, 347–370 (1986)

    Google Scholar 

  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. Van den Berg, J., Kesten H.: Inequalities with applications to percolation and reliability. J. Appl. Probab. 22, 556–599 (1985)

    Google Scholar 

  11. Liggett T.M.: Interacting Particle Systems. Berlin Heidelberg New York: Springer 1985

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Minnesota, 55455, Mineapolis, MN, USA

    Maury Bramson

  2. Department of Mathematics, Syracuse University, 13210, Syracuse, NY, USA

    J. Theodore Cox

  3. Department of Mathematics, University of Wisconsin, 53706, Madison, WI, USA

    David Griffeath

Authors
  1. Maury Bramson
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  2. J. Theodore Cox
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  3. David Griffeath
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Additional information

Dedicated to Frank Spitzer on his 60th birthday

Partially supported by the National Science Foundation under Grant DMS-831080

Partially supported by the National Science Foundation under Grant DMS-841317

Partially supported by the National Science Foundation under Grant DMS-830549

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Cite this article

Bramson, M., Theodore Cox, J. & Griffeath, D. Occupation time large deviations of the voter model. Probab. Th. Rel. Fields 77, 401–413 (1988). https://doi.org/10.1007/BF00319297

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  • Received: 02 February 1986

  • Revised: 19 October 1987

  • Issue Date: March 1988

  • DOI: https://doi.org/10.1007/BF00319297

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Keywords

  • Lower Bound
  • Stochastic Process
  • Decay Rate
  • Probability Theory
  • High Dimension
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