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The continuum random cluster model is a Gibbs modification of the standard Boolean model with intensity z > 0 and law of radii Q. The formal unnormalised density is given by qNcc, where q is a fixed parameter and Ncc is the number of connected components in the random structure. We prove for a large class of parameters that percolation occurs for large enough z and does not occur for small enough z. We provide an application to the phase transition of the Widom–Rowlinson model with random radii. Our main tools are stochastic domination properties, a detailed study of the interaction of the model, and a Fortuin–Kasteleyn representation.
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