Uniqueness of the infinite cluster for stationary Gibbs states (Q909364)

It has been shown for the Bernoulli model of percolation on \(Z^ d\) (where the probability of each spin being up equals p independent of the other spins) that, with probability one, the number N of infinite clusters can only assume the values 0 or 1 [cf., e.g., the author, \textit{G. Grimmett} and \textit{L. Russo}, Commun. Math. Phys. 114, 549-552 (1988; Zbl 0649.60104)]. In the present paper the number N of infinite clusters is studied for percolation in Gibbs models on \(Z^ d\) where two nearest-neighbour sites are connected if the spins in these sites are up. The author shows that also in the case of stationary Gibbs states where the interaction has finite range or decreases sufficiently rapidly, N assumes only the values 0 or 1 with probability 1. This generalizes, e.g., some results of \textit{A. Coniglio}, \textit{C. R. Nappi}, \textit{F. Peruggi} and \textit{L. Russo} for some Gibbs models in dimension 2 [ibid. 51, 315-323 (1976)].