Electronic Journal of Probability

Infinite volume continuum random cluster model

David Dereudre and Pierre Houdebert

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The continuum random cluster model is defined as a Gibbs modification of the stationary Boolean model in $\mathbb{R}^d$ with intensity $z \gt 0$ and the law of radii $Q$. The formal unormalized density is given by $q^{N_{cc}}$ where $q \gt 0$ is a fixed parameter and $N_{cc}$ the number of connected components in the random germ-grain structure. In this paper we prove the existence of the model in the infinite volume regime for a large class of parameters including the case $q \lt 1$ or distributions $Q$ without compact support. In the extreme setting of non integrable radii (i.e. $\int R^d Q(dR)=\infty$) and $q$ is an integer larger than 1, we prove that for $z$ small enough the continuum random cluster model is not unique; two different probability measures solve the DLR equations. We conjecture that the uniqueness is recovered for $z$ large enough which would provide a phase transition result. Heuristic arguments are given. Our main tools are the compactness of level sets of the specific entropy, a fine study of the quasi locality of the Gibbs kernels and a Fortuin-Kasteleyn representation via Widom-Rowlinson models with random radii.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 125, 24 pp.

Received: 10 July 2015
Accepted: 29 November 2015
First available in Project Euclid: 4 June 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G10: Stationary processes 60G55: Point processes 60G57: Random measures 60G60: Random fields

Gibbs point process phase transition specific entropy Boolean model Widom-Rowlinson model Fortuin-Kasteleyn representation

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Dereudre, David; Houdebert, Pierre. Infinite volume continuum random cluster model. Electron. J. Probab. 20 (2015), paper no. 125, 24 pp. doi:10.1214/EJP.v20-4718. https://projecteuclid.org/euclid.ejp/1465067231

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