Infinite Volume Continuum Random Cluster Model

2015/07/10 by Dereudre, David, Houdebert, Pierre · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1507.02891

Abstract

The continuum random cluster model is defined as a Gibbs modification of the stationary Boolean model in ℝ^d with intensity z>0 and the law of radii Q. The formal unormalized density is given by q^N_cc where q>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<1 or distributions Q without compact support. In the extreme setting of non integrable radii (i.e. ∫ R^d Q(dR)=∞) 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.

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• Phase transitions for the Widom--Rowlinson model in random environments

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