Infinite Volume Continuum Random Cluster Model

TL;DR

This paper establishes the existence of the infinite volume continuum random cluster model for a broad parameter range, explores non-uniqueness in certain regimes, and conjectures phase transition phenomena.

Contribution

It proves the existence of the continuum random cluster model in infinite volume for various parameters, including non-compact radii distributions, and analyzes non-uniqueness and phase transitions.

Findings

Existence of the model in infinite volume for large classes of parameters.

Non-uniqueness of Gibbs measures for certain radii distributions and parameters.

Heuristic conjecture on phase transition at large intensity.

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.