Abstract
We study the continuum Widom-Rowlinson model of interpenetrating spheres. Using a new geometric representation for this system we provide a simple percolation-based proof of the phase transition. We also use this representation to formulate the problem, and prove the existence of an interfacial tension between coexisting phases. Finally, we ascribe geometric (i.e. probabilistic) significance to the correlation functions which allows us to prove the existence of a sharp correlation length in the single-phase regime.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[A] Aizenman, M.: Geometric Analysis of Φ4 Fields and Ising Models I, II. Commun. Math. Phys.86, 1–48 (1982)
[ACCN] Aizenman, M., Chayes, J.T., Chayes, L., Newman, C.M.: Discontinuity of the Magnetization in One-Dimensional 1/|x−y|2 Ising and Potts Models. J. Stat. Phys.50, 1–40 (1988)
[CGLM] Cassandro, M., Gallavotti, G., Lebowitz, J.L., Monroe, J.L.: Existence and Uniqueness of Equilibrium States for Some Spin and Continuum systems. Commun. Math. Phys.32, 153–165 (1973)
[CK] Chayes, L., Klein, D.: A Generalization of Poisson Convergence to “Gibbs Convergence” with Applications to Statistical Mechanics. Helv. Phys. Acta67, 30–42 (1994)
[FK] Fortuin, C.M., Kasteleyn, P.W.: On the Random Cluster Model I. Introduction and Relation to Other Models. Physica57, 536–564 (1972)
[FKG] Fortuin, C.M., Kasteleyn, P.W., Ginibre, J.: Correlation Inequalities on Some Partially Ordered Sets. Commun. Math. Phys.22, 89–103 (1971)
[Ge] Georgii, H.-O.: Private communication, unpublished
[GLM] Giacomin, G., Lebowitz, J.L., Maes, C.: Agreement Percolation and Phase Coexistence in Some Gibbs Systems. Preprint
[Gr] Grimmett, G.: Percolation, Heidelberg, New York: Springer, Berlin, 1989
[GS] Givens, J.A., Stell, G.: The Kirkwood-Salsburg Equation for Random Continuum Percolation. J. Stat. Phys.,59, 981–1018 (1980)
[H] Harris, T.E.: A Lower Bound for the Critical Probability in a Certain Percolation Process. Proc. Camb. Phil. Soc.56, 13–20 (1960)
[Ho] Holley, R.: Remarks on the FKG Inequalities. Commun. Math. Phys.36, 227–231 (1974)
[J] Janson, S.: Bounds on the Distribution of Extremal Values of a Scanning Process. Stoch. Proc. Appl.318, 313–328 (1984)
[K] Klein, W.: Potts-Model Formulation of Continuum Percolation. Phys. Rev. B26, 2677–2678 (1982)
[L] Liggett, T.M.: Interacting Particle Systems. Heidelberg, New York: Springer, Berlin, 1985
[LG] Lebowitz, J.L., Gallavotti, G.: Phase Transitions in Binary Lattice Gases. J. Math. Phys.12, 1229–1233 (1971)
[LM] Lebowitz, J.L., Monroe, J.L.: Inequalities for Higher Order Ising Spins, and for Continuum Fluids. Commun. Math. Phys.28, 301–311 (1972)
[LL] Lebowitz, J.L., Lieb, E.H.: Phase Transitions in a Continuum Classical System with Finite Interactions. Phys. Lett.39A, 98–100 (1972)
[P] Preston, C.: Spatial Birth-and-Death Processes. Bull. Inst. Inter. Stat.46 (2), 371–391 (1975)
[R] Ruelle, D.: Existence of a Phase Transition in a Continuous Classical System. Phys. Rev. Lett.27, 1040–1041 (1971)
[Ru] Russo, L.: An Approximate Zero-One Law. Zeitschrift für Wahrscheinlichkeitstheorie and Ihre Grenzgebiete61, 129–139 (1982)
[WR] Widom, B., Rowlinson, J.S.: New Model for the Study of Liquid-Vapor Phase Transitions. J. Chem. Phys.52, 1670–1684 (1970)
Author information
Authors and Affiliations
Additional information
Communicated by D. Brydges
Partly supported by the grants GAČR 202/93/0499, GAUK 376, NSF-DMS 91-04487, and NSF-DMS 93-02023
Rights and permissions
About this article
Cite this article
Chayes, J.T., Chayes, L. & Kotecký, R. The analysis of the Widom-Rowlinson model by stochastic geometric methods. Commun.Math. Phys. 172, 551–569 (1995). https://doi.org/10.1007/BF02101808
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02101808