Percolation results for the continuum random cluster model. (20th March 2018) - Houdebert, Pierre
Abstract: 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 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 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.