A Monotonicity Result for Hard-core and Widom-Rowlinson Models on Certain $d$-dimensional Lattices

Olle Häggström (Chalmers University of Technology and Goteborg University)


For each $d\geq 2$, we give examples of $d$-dimensional periodic lattices on which the hard-core and Widom-Rowlinson models exhibit a phase transition which is monotonic, in the sense that there exists a critical value $\lambda_c$ for the activity parameter $\lambda$, such that there is a unique Gibbs measure (resp. multiple Gibbs measures) whenever $\lambda$ is less than $\lambda_c$ (resp. $\lambda$ greater than $\lambda_c$). This contrasts with earlier examples of such lattices, where the phase transition failed to be monotonic. The case of the cubic lattice $Z^d$ remains an open problem.

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Pages: 67-78

Publication Date: February 2, 2002

DOI: 10.1214/ECP.v7-1048


  1. G. Brightwell, O. Häggström, and P. Winkler (1999), Nonmonotonic behavior in hard-core and Widom--Rowlinson models, J. Statist. Phys. 94, 415-435. Math. Review 2000c:82016
  2. H.-O. Georgii, O. Häggström, and C. Maes (2001), The random geometry of equilibrium phases, Phase Transitions and Critical Phenomena, Volume 18 (C. Domb and J.L. Lebowitz, eds), pp 1-142, Academic Press, London.
  3. G.R. Grimmett (1999), Percolation (2nd edition), Springer, New York. Math. Review 2001a:60114
  4. O. Häggström (2000), Markov random fields and percolation on general graphs, Adv. Appl. Probab. 32, 39-66. Math. Review 2001g:60246
  5. F.P. Kelly (1985), Stochastic models of computer communication systems J. Roy. Statist. Soc. B 47, 379-395. Math. Review 87m:60212a
  6. J.C. Wheeler and B. Widom (1970) Phase equilibrium and critical behavior in a two-component Bethe-lattice gas or three-component Bethe-lattice solution, J. Chem. Phys. 52, 5334-5343.

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