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Consolidation rates for two interacting systems in the plane
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  • Published: November 1986

Consolidation rates for two interacting systems in the plane

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

Probability Theory and Related Fields volume 73, pages 613–625 (1986)Cite this article

  • 103 Accesses

  • 11 Citations

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Summary

This paper is a sequel of a paper of Cox and Griffeath “diffusive clustering in the two dimensional voter model”. We continue our study of the voter model and coalescing random walks on the two dimensional integer lattice. Some exact asymptotics concerning the rate of clustering in the former process and the coalescence rate of the latter are derived. We use these results to prove a limit law, announced in that earlier paper, concerning the size of the largest square centered at the origin which is of solid color at a large time t.

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References

  1. Bramson, M., Griffeath, D.: Asymptotics for interacting particle systems on \(\mathbb{Z}^{\text{d}} \). Z. Wahrscheinlichkeitstheor. Verw. Geb. 45, 183–196 (1980)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  3. Griffeath, D.: Additive and cancellative interacting particle systems. Lecture Notes Math. 724. Berlin-Heidelberg-New York: Springer 1979

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  4. Holley, R., Stroock, D.: Central limit phenomena of various interacting systems. Ann. Math. 110, 333–393 (1979)

    MathSciNet  Google Scholar 

  5. Liggett, T.M.: Interacting particle systems. Berlin-Heidelberg-New York: Springer 1985

    Google Scholar 

  6. Tavaré, S.: Line-of-descent and geneological processes, and their applications in population genetics models. Theor. Popul. Biol. 26, 119–164 (1984)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Minnesota, 55455, Minneapolis, 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

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., Cox, J.T. & Griffeath, D. Consolidation rates for two interacting systems in the plane. Probab. Th. Rel. Fields 73, 613–625 (1986). https://doi.org/10.1007/BF00324856

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  • Received: 17 July 1985

  • Issue Date: November 1986

  • DOI: https://doi.org/10.1007/BF00324856

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Keywords

  • Color
  • Stochastic Process
  • Probability Theory
  • Large Time
  • Mathematical Biology
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