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Reversible growth models on symmetric sets. (English) Zbl 0653.60094
Probabilistic methods in mathematical physics, Proc. Taniguchi Int. Symp., Katata and Kyoto/Jap. 1985, 275-301 (1987).
[For the entire collection see Zbl 0633.00021.]
In Probab. Theory, Relat. Fields 74, 505-528 (1987; Zbl 0589.60081), the author expressed the survival probability of reversible growth models on \({\mathbb{Z}}\) by a variational formula derived through an application of the Dirichlet principle. The present paper generalizes this result to general symmetric sets, e.g. \({\mathbb{Z}}^ d.\) It gives upper and lower bounds for the survival probability in order to calculate the critical value and exponent of a birth rate parameter \(\lambda\).
The models show that the one-dimensional situation can be completely different from higher-dimensional ones.
Reviewer: Th.Eisele

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)