arXiv 1507.028912015-07math.PR

无限体积连续随机簇模型

Infinite Volume Continuum Random Cluster Model

David Dereudre, Pierre Houdebert

摘要：

连续随机簇模型被定义为在$\mathbb{R}^d$中具有强度$z>0$和半径定律$Q$的固定布尔模型的Gibbs修改。正式的未正规化密度由$q^{N_{cc}}$给出，其中$q>0$是固定参数，$N_{cc}$是随机胚粒结构中的连通分量的数目。在本文中，我们证明了该模型的存在性，在无限体积区的一大类参数，包括情况$q<1$或分布$Q$没有紧支持。在非可积半径（即$\int R^d Q(dR)=\infty$）和$q$是大于1的整数的极端情况下，我们证明了对于足够小的$z$，连续随机簇模型不是唯一的;两种不同的概率测度解DLR方程.我们推测，恢复的唯一性$z$足够大，这将提供一个相变的结果。启发式参数。我们的主要工具是紧的水平集的特定熵，一个很好的研究的准局部性的吉布斯核和Fortuin-Kasteleyn表示通过Widom-Rowlinson模型与随机半径。

英文摘要：

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.

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