We generalise disagreement percolation to Gibbs point processes of balls with varying radii. This allows to establish the uniqueness of the Gibbs measure and exponential decay of pair correlations in the low activity regime by comparison with a sub-critical Boolean model. Applications to the Continuum Random Cluster model and the Quermass-interaction model are presented. At the core of our proof lies an explicit dependent thinning from a Poisson point process to a dominated Gibbs point process.

The continuum random cluster model is defined as a Gibbs modification of the stationary Boolean model in R d with intensity z > 0 and the law of radii Q. The formal unormalized density is given by q Ncc 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. R d Q(dR) = ∞) 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.

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 unnormalized density is given by q Ncc 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.

The Widom-Rowlinson model (or the Area-interaction model) is a Gibbs point process in R d with the formal Hamiltonian defined as the volume of ∪ x∈ω B 1 (x), where ω is a locally finite configuration of points and B 1 (x) denotes the unit closed ball centred at x. The model is also tuned by two other parameters: the activity z > 0 related to the intensity of the process and the inverse temperature β ≥ 0 related to the strength of the interaction. In the present paper we investigate the phase transition of the model in the point of view of percolation theory and the liquid-gas transition. First, considering the graph connecting points with distance smaller than 2r > 0, we show that for any β ≥ 0, there exists 0 < z a c (β, r) < +∞ such that an exponential decay of connectivity at distance n occurs in the subcritical phase (i.e. z < z a c (β, r)) and a linear lower bound of the connection at infinity holds in the supercritical case (i.e. z > z a c (β, r)). These results are in the spirit of recent works using the theory of randomised tree algorithms [7,9,8]. Secondly we study a standard liquid-gas phase transition related to the uniqueness/non-uniqueness of Gibbs states depending on the parameters z, β. Old results [22,24] claim that a non-uniqueness regime occurs for z = β large enough and it is conjectured that the uniqueness should hold outside such an half line (z = β ≥ β c > 0). We solve partially this conjecture in any dimension by showing that for β large enough the non-uniqueness holds if and only if z = β. We show also that this critical value z = β corresponds to the percolation threshold z a c (β, r) = β for β large enough, providing a straight connection between these two notions of phase transition.