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Percolation results for the continuum random cluster model

Part of: Equilibrium statistical mechanics Special processes Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  20 March 2018

Pierre Houdebert
Pierre Houdebert*
Affiliation:
Université Lille 1
*
* Current address: Centre de Mathématiques et Informatique (CMI), Aix-Marseille Université, Technopôle Château-Gombert, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France. Email address: [email protected]
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Abstract

The continuum random cluster model is a Gibbs modification of the standard Boolean model with intensity z > 0 and law of radii Q. The formal unnormalised density is given by q N cc , where q is a fixed parameter and N cc is the number of connected components in the random structure. We prove for a large class of parameters that percolation occurs for large enough z and does not occur for small enough z. We provide an application to the phase transition of the Widom–Rowlinson model with random radii. Our main tools are stochastic domination properties, a detailed study of the interaction of the model, and a Fortuin–Kasteleyn representation.

Keywords

Gibbs point processWidom–Rowlinson modelstochastic comparisonFortuin–Kasteleyn representation

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G10: Stationary processes 60G55: Point processes 60G57: Random measures 60G60: Random fields 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B21: Continuum models (systems of particles, etc.) 82B26: Phase transitions (general) 82B43: Percolation
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 1 , March 2018 , pp. 231 - 244
Copyright
Copyright © Applied Probability Trust 2018 

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