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.
References
Durrett, R.: Oriented percolation in two dimensions. Ann. Probab. 12, 999–1040 (1984)
Gray, L., Griffeath, D.: A stability criterion for attractive nearest neighbor spin systems on 298-1. Ann. Probab. 10, 67–85 (1982)
Liggett, T.M.: Interacting particle systems. New York Berlin Heidelberg: Springer, 1985
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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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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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DOI: https://doi.org/10.1007/BF00356107
Keywords
- Stochastic Process
- Probability Theory
- Mathematical Biology
- Small Perturbation
- Stability Criterion