The analysis of the Widom-Rowlinson model by stochastic geometric methods

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.

[1]  S. Janson.  Bounds on the distributions of extremal values of a scanning process , 1984 .

[2]  E. Lieb,et al.  Phase transition in a continuum classical system with finite interactions , 1972 .

[3]  G. Giacomin,et al.  Agreement percolation and phase coexistence in some Gibbs systems , 1995 .

[4]  L. Russo.  An approximate zero-one law , 1982 .

[5]  M. Aizenman,et al.  Geometric analysis of ϕ 4 fields and Ising models , 1982 .

[6]  W. Klein.  Potts-model formulation of continuum percolation , 1982 .

[7]  J. Lebowitz,et al.  Existence and uniqueness of equilibrium states for some spin and continuum systems , 1973 .

[8]  Michael Aizenman,et al.  Geometric analysis of φ4 fields and Ising models. Parts I and II , 1982 .

[9]  J. Given,et al.  The kirkwood-salsburg equations for random continuum percolation , 1990 .

[10]  T. E. Harris.  A lower bound for the critical probability in a certain percolation process , 1960, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  C. Preston.  Spatial birth and death processes , 1975, Advances in Applied Probability.

[12]  J. Lebowitz,et al.  Inequalities for higher order Ising spins and for continuum fluids , 1972 .

[13]  J. Chayes,et al.  Discontinuity of the magnetization in one-dimensional 1/¦x−y¦2 Ising and Potts models , 1988 .

[14]  C. Fortuin,et al.  On the random-cluster model: I. Introduction and relation to other models , 1972 .

[15]  David Ruelle,et al.  A phase transition in a continuous classical system , 1971 .

[16]  C. Fortuin,et al.  Correlation inequalities on some partially ordered sets , 1971 .

[17]  R. Holley.  Remarks on the FKG inequalities , 1974 .

[18]  A generalization of Poisson convergence to Gibbs convergence with applications to statistical mechanics , 1994 .

[19]  J. Lebowitz,et al.  Phase Transitions in Binary Lattice Gases , 1971 .

[20]  John S. Rowlinson,et al.  New Model for the Study of Liquid–Vapor Phase Transitions , 1970 .

[21]  T. Liggett.  Interacting Particle Systems , 1985 .