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Probability Theory and Related Fields

, Volume 80, Issue 2, pp 293–298 | Cite as

A simple proof of the stability criterion of Gray and Griffeath

  • Maury Bramson
  • Rick Durrett
Article

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.

Keywords

Stochastic Process Probability Theory Mathematical Biology Small Perturbation Stability Criterion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Durrett, R.: Oriented percolation in two dimensions. Ann. Probab. 12, 999–1040 (1984)zbMATHMathSciNetGoogle Scholar
  2. Gray, L., Griffeath, D.: A stability criterion for attractive nearest neighbor spin systems on 298-1. Ann. Probab. 10, 67–85 (1982)MathSciNetGoogle Scholar
  3. Liggett, T.M.: Interacting particle systems. New York Berlin Heidelberg: Springer, 1985Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Maury Bramson
    • 1
  • Rick Durrett
    • 2
  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA
  2. 2.Department of MathematicsCornell UniversityIthacaUSA

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