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


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.


Stochastic Process Probability Theory Mathematical Biology Small Perturbation Stability Criterion 
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  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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