Abstract
A uniqueness condition for Gibbs measures is given. This condition is stated in terms of (absence of) a certain type of percolation involving two independent realisations. This result can be applied in certain concrete situations by comparison with “ordinary” percolation. In this way we prove that the Ising antiferromagnet on a square lattice has a unique Gibbs measure if β(4−|h⥻)<1/2ln(P c /(1−P c )), whereh denotes the external magnetic field, β the inverse temperature, andP c the critical probability for site percolation on that lattice. SinceP c is larger than 1/2, this extends a result by Dobrushin, Kolafa and Shlosman (whose proof was computer-assisted).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aizenman, M.: Translation invariance and instability of phase coexistence in the two-dimensional Ising system. Commun. Math. Phys.73, 83–94 (1980)
Berg, J., van den, Steif, J.E.: On the hard-core lattice gas model, percolation and certain loss networks. Preprint (1992)
Berg, J., van den, Maes, C.: Preprint (1992)
Blöte, H.W.J., Wu, X.-N.: Accurate determination of the critical line of the square Ising antiferromagnet in a field. J. Phys. A: Math. Gen.23, L627-L631 (1990)
Broadbent, S.R., Hammersley, J.M.: Percolation processes. I. Crystals and mazes. Proceedings of the Cambridge Philosophical Society53, 629–641 (1957)
Dobrushin, R.L.: The description of a random field by means of conditional probabilities and conditions of its regularity. Theor. Prob. Appl.13, 197–224 (1968a)
Dobrushin, R.L.: The problem of uniqueness of a Gibbs random field and the problem of phase transition. Funct. Anal. Appl.2, 302–312 (1968b)
Dobrushin, R.L., Shlosman, S.B.: Constructive criterion for the uniqueness of a Gibbs field. In: Fritz, J., Jaffe, A., Szász, D. (eds.), Statistical mechanics and dynamical systems. Boston: Birkhäuser 1985, pp. 371–403
Dobrushin, R.L., Kolafa, J., Shlosman, S.B.: Phase diagram of the two-dimensional Ising antiferromagnetic (computer-assisted proof). Commun. Math. Phys.102, 89–103 (1985)
Fisher, M.E.: Critical probabilities for cluster size and percolation problems. J. Math. Phys.2, 620–627 (1961)
Georgii, H.-O.: Gibbs measures and phase transitions. Berlin, New York: de Gruyter 1988
Grimmett, G.R.: Percolation. Berlin, Heidelberg, New York: Springer 1989
Grimmett, G.R., Marstrand, J.M.: The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A430, 439–457 (1990)
Hammersley, J.M.: Comparison of atom and bond percolation. J. Math. Phys.2, 728–733 (1961)
Harris, T.E.: A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Phil. Soc.56, 13–20 (1960)
Higuchi, Y.: Coexistence of the infinite (*) clusters: a remark on the square lattice site percolation. ZfW61, 75–81 (1982)
Kesten, H.: Percolation theory for mathematicians. Boston: Birkhäuser 1982
Kindermann, R., Snell, J.L.: Markov random fields and their applications. Contemporary Mathematics, Vol. 1. Providence, R.I.: Amer. Math. Soc. 1980
Lebowitz, J.L.: GHS and other inequalities. Commun. Math. Phys.35, 87–92 (1974)
Menshikov, M.V., Pelikh, K.D.: Matematicheskie Zametki46, 38–47 (1989)
Percus, J.: Correlation inequalities for Ising spin lattices. Commun. Math. Phys.40, 283–308 (1975)
Prum, B., Fort, J.C.: Stochastic processes on a lattice and Gibbs measures. Dordrecht, Boston, London: Kluwer 1991
Russo, L.: The infinite cluster method in the two-dimensional Ising model. Commun. Math. Phys.67, 251–266 (1979)
Toth, B.: A lower bound for the critical probability of the square lattice site percolation. ZfW69, 19–22 (1985)
Author information
Authors and Affiliations
Additional information
Communicated by M. Aizenman
The research which led to this paper started while the author was at Cornell University, partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University
Rights and permissions
About this article
Cite this article
van den Berg, J. A uniqueness condition for Gibbs measures, with application to the 2-dimensional Ising antiferromagnet. Commun.Math. Phys. 152, 161–166 (1993). https://doi.org/10.1007/BF02097061
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02097061