Abstract
We give two examples of nonmonotonic behavior in symmetric systems exhibiting more than one critical point at which spontanoous symmetry breaking appears or disappears. The two systems are the hard-core model and the Widom–Rowlinson model, and both examples take place on a variation of the Cayley tree (Bethe lattice) devised by Schonmann and Tanaka. We obtain similar, though less constructive, examples of nonmonotonicity via certain local modifications of any graph, e.g., the square lattice, which is known to have a critical point for either model. En route we discuss the critical behavior of the Widom–Rowlinson model on the ordinary Cayley tree. Some results about monotonicity of the phase transition phenomenon relative to graph structure are also given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
I. Benjamini and O. Schramm, Percolation beyond Z d: many questions and a few answers, Electr. Commun. Probab. 1:71–82 (1996).
J. Van den Berg, A uniqueness condition for Gibbs measures with application to the 2–dimensional Ising antiferromagnet, Commun. Math. Phys. 152:161–166 (1993).
J. Van den Berg and A. Ermakov, A new lower bound for the critical probability of site percolation on the square lattice, Random Structures Algorithms 8:199–212 (1996).
J. Van den Berg and J. Steif, Percolation and the hard core lattice gas model, Stoch. Proc. Appl. 49:179–197 (1994).
G. Brightwell, O. Häggström, and P. Winkler, Nonmonotonic behavior in hard-core and Widom-Rowlinson models, CDAM Research Report LSE-CDAM-98–13.
G. Brightwell and P. Winkler, Graph homomorphisms and phase transitions, J. Comb. Theory B, to appear (1998).
G. Brightwell and P. Winkler, Gibbs measures and dismantlable graphs, CDAM Research Report LSE-CDAM-97–17.
J. T. Chayes, L. Chayes, and R. Kotecký, The analysis of the Widom-Rowlinson model by stochastic geometric methods, Commun. Math. Phys. 172:551–569 (1995).
R. L. Dobrushin, The problem of uniqueness of a Gibbs random field and the problem of phase transition, Funct. Anal. Appl. 2:302–312 (1968).
H.-O. Georgii, Gibbs Measures and Phase Transitions (de Gruyter, New York, 1988).
G. Giacomin, J. L. Lebowitz, and C. Maes, Agreement percolation and phase coexistence in some Gibbs systems, J. Statist. Phys. 80:1379–1403 (1995).
G. Grimmett, Percolation (Springer, New York, 1989).
O. Häggström, Ergodicity of the hard core model on Z 2 with parity-dependent activities, Ark. Mat. 35:171–184 (1997).
O. Häggström, Random-cluster representations in the study of phase transitions, Markov Proc. Relat. Fields 4:275–321 (1998).
O. Häggström, Random-cluster analysis of a class of binary lattice gases, J. Statist. Phys. 91:47–74 (1998).
F. P. Kelly, Stochastic models of computer communication systems, J. Roy. Statist. Soc. B 47:379–395 (1985).
J. L. Lebowitz and G. Gallavotti, Phase transitions in binary lattice gases, J. Math. Phys. 12:1129–1133 (1971).
J. L. Lebowitz, A. Mazel, P. Nielaba, and L. Šamaj, Ordering and demixing transitions in multicomponent Widom-Rowlinson models, Phys. Rev. E 52:5985–5996 (1995).
T. M. Liggett, Interacting Particle Systems (Springer, New York, 1985).
R. Schonmann and N. Tanaka, Lack of monotonicity in ferromagnetic Ising model phase diagrams, Ann. Appl. Probab. 8:234–245 (1998).
H. Tanemura, A system of infinitely many mutually reflecting Brownian balls in R d, Probab. Th. Relat. Fields 104:399–426 (1996).
J. C. Wheeler and B. Widom, Phase equilibrium and critical behavior in a two-component Bethe-lattice gas or three-component Bethe-lattice solution, J. Chem. Phys. 52:5334–5343 (1970).
B. Widom and J. S. Rowlinson, New model for the study of liquid-vapor phase transition, J. Chem. Phys. 52:1670–1684 (1970).
J. C. Wierman, Substitution method critical probability bounds for the square lattice site percolation model, Comb. Prob. Computing 4:181–188 (1995).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Brightwell, G.R., Häggström, O. & Winkler, P. Nonmonotonic Behavior in Hard-Core and Widom–Rowlinson Models. Journal of Statistical Physics 94, 415–435 (1999). https://doi.org/10.1023/A:1004592103315
Issue Date:
DOI: https://doi.org/10.1023/A:1004592103315