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- W1868377977 abstract "The continuum random cluster model is defined as a Gibbs modification of the stationary Boolean model in $mathbb{R}^d$ 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. $int R^d Q(dR)=infty$) 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." @default.
- W1868377977 created "2016-06-24" @default.
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- W1868377977 date "2015-07-10" @default.
- W1868377977 modified "2023-09-27" @default.
- W1868377977 title "Infinite Volume Continuum Random Cluster Model" @default.
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