Summary
The random-cluster model on a homogeneous tree is defined and studied. It is shown that for 1≦q≦2, the percolation probability in the maximal random-cluster measure is continuous inp, while forq>2 it has a discontinuity at the critical valuep=p c (q). It is also shown that forq>2, there is nonuniqueness of random-cluster measures for an entire interval of values ofp. The latter result is in sharp contrast to what happens on the integer lattice Zd.
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References
Aizenman, M., Chayes, J.T., Chayes, L., Newman, C.M.: Discontinuity of the magnetization in one-dimensional 252-1 Ising and Potts models. J. Stat. Phys.50, 1–40 (1988)
Benjamini, I., Peres, Y.: Markov chains indexed by trees. Ann. Probab.22, 219–243 (1994)
Bleher, P.M., Ruiz, J., Zagrebnov, V.A.: On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Stat. Phys.79, 473–482 (1995)
Bollobás, B., Grimmett, G., Janson, S.: The random-cluster model on the complete graph. Probab. Theory Relat. Fields104, 383–417 (1996)
Burton, R., Keane, M.: Density and uniqueness in percolation. Comm. Math. Phys.121, 501–505 (1989)
Chayes, J.T., Chayes, L., Sethna, J.P., Thouless, D.J.: A mean field spin glass with short range interactions. Comm. Math. Phys.106, 41–89 (1986)
Edwards, R.G., Sokal, A.D.: Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D38, 2009–2012 (1988)
Fortuin, C.M.: On the random-cluster model. II. The percolation model. Physica58, 393–418 (1972)
Fortuin, C.M.: On the random-cluster model. III. The simple random-cluster process. Physica59, 545–570 (1972)
Fortuin, C.M., Kasteleyn, P.W.: On the random-cluster model. I. Introduction and relation to other models. Physica57, 536–564 (1972)
Georgii, H.-O.: Gibbs Measures and Phase Transitions. W. de Gruyter, New York (1988)
Grimmett, G.: Percolation. Springer, New York (1989)
Grimmett, G.: Percolative problems. Probability and Phase Transition. Kluwer, Dordrecht, (1994), pp. 69–86
Grimmett, G.: The stochastic random-cluster process, and the uniqueness of randomcluster measures. Ann. Probab. (in press)
Häggström, O.: Random-cluster measures and uniform spanning trees. Stoch. Proc. Appl. (in press)
Higuchi, Y.: Remarks on the limiting Gibbs states on a (d+1) tree. Publ. RIMS13, 335–348 (1977)
Lebowitz, J., Martin-Löf, A.: On the uniqueness of the equilibrium state for Ising spin systems. Comm. Math. Phys.25, 276–282 (1972)
Lyons, R.: The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys.125, 337–353 (1989)
Lyons, R.: Random walks and percolation on trees. Ann. Probab.18, 931–958 (1990)
Spitzer, F.: Markov random fields on an infinite tree. Ann. Probab.3, 387–398 (1975)
Zachary, S.: Countable state space Markov random fields and Markov chains on trees. Ann. Probab.11, 894–903 (1983)
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Research partially supported by a grant from the Royal Swedish Academy of Sciences