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A simple proof of the stability criterion of Gray and Griffeath
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  • Published: December 1988

A simple proof of the stability criterion of Gray and Griffeath

  • Maury Bramson1 &
  • Rick Durrett2 

Probability Theory and Related Fields volume 80, pages 293–298 (1988)Cite this article

  • 128 Accesses

  • 34 Citations

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Summary

Gray and Griffeath studied attractive nearest neighbor spin systems on the integers having “all 0's” and “all 1's” as traps. Using the contour method, they established a necessary and sufficient condition for the stability of the “all 1's” equilibrium under small perturbations. In this paper we use a renormalized site construction to give a much simpler proof. Our new approach can be used in many situations as a substitute for the contour method.

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References

  • Durrett, R.: Oriented percolation in two dimensions. Ann. Probab. 12, 999–1040 (1984)

    MATH  MathSciNet  Google Scholar 

  • Gray, L., Griffeath, D.: A stability criterion for attractive nearest neighbor spin systems on 298-1. Ann. Probab. 10, 67–85 (1982)

    MathSciNet  Google Scholar 

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

    Google Scholar 

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Author information

Authors and Affiliations

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

    Maury Bramson

  2. Department of Mathematics, Cornell University, White Hall, 14853, Ithaca, NY, USA

    Rick Durrett

Authors
  1. Maury Bramson
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  2. Rick Durrett
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Additional information

Partially supported by a grant from the National Science Foundation

Partially supported by the Army Research Office through the Mathematical Sciences Institute at Cornell University

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Cite this article

Bramson, M., Durrett, R. A simple proof of the stability criterion of Gray and Griffeath. Probab. Th. Rel. Fields 80, 293–298 (1988). https://doi.org/10.1007/BF00356107

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  • Received: 21 April 1988

  • Revised: 04 July 1988

  • Issue Date: December 1988

  • DOI: https://doi.org/10.1007/BF00356107

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Keywords

  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Small Perturbation
  • Stability Criterion
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