Percolation results for the Continuum Random Cluster Model

TL;DR

This paper investigates the percolation phase transition in the continuum random cluster model, establishing conditions for the occurrence and absence of percolation, and applies findings to the Widom-Rowlinson model with random radii.

Contribution

It provides new results on percolation thresholds in the continuum random cluster model using stochastic domination and Fortuin-Kasteleyn representation.

Findings

Percolation occurs for large enough intensity z.

Percolation does not occur for small enough z.

Application to phase transition in the Widom-Rowlinson model.

Abstract

The continuum random cluster model is a Gibbs modification of the standard boolean model of intensity $z > 0$ and law of radii $Q$. The formal unormalized density is given by $q^{N_{cc}}$ where $q$ is a fixed parameter and $N_{cc}$ is the number of connected components in the random structure. We prove for a large class of parameters that percolation occurs for $z$ large enough and does not occur for $z$ small enough. An application to the phase transition of the Widom-Rowlinson model with random radii is given. Our main tools are stochastic domination properties, a fine study of the interaction of the model and a Fortuin-Kasteleyn representation.