Percolation results for the continuum random cluster model

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 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.

Disagreement percolation for Gibbs ball models

“…Taking the derivative along the ordered space (∆, ) yields (16). On the lefthand side of (19) we apply a one-sided version of the Lebesgue differentiation theorem [2, Thm 5.6.2] to extract the integrand as the Q ⋆ -a.e.…”

“…derivative. There is an additional minus sign in (16), because differentiation of the left-hand side of (19) proceeds in decreasing direction in , reverse to the direction used the common direction used in the derivative (13). Because Q ⋆ is diffuse, the negligible change of the left interval border from closed to open in the left-hand side of (18b) does not matter.…”

Phase Transition for Continuum Widom–Rowlinson Model with Random Radii

“…, z) large enough, the set W R(z, Q) contains q distinct extremal ordered phases. These results derive from a coupling result in [14], a standard Fortuin-Kasteleyn representation and a percolation result developed in [16]. In conclusion, in the integrable symmetric setting, W R(z, Q) exhibits a standard phase transition similar to the deterministic radii case.…”

Phase transition for the non-symmetric Continuum Potts model

“…This model was first introduced in [18] and then used in [13] and [3] to prove phase transition, by providing an uniform bound (with respect to the finite volume box Λ) of the percolative probability that the boundary is connected to the origin. The Continuum random cluster model was also studied on its own in [7,17].…”

“…In the 1990' Chayes, Chayes & Kotecký [3] and Georgii & Häggström [13] generalized for continuum models the idea of the Fortuin-Kasteleyn representation [10] introduced for the lattice Ising and Potts models, and proved phase transition results respectively for the symmetric Widom-Rowlinson model and for the continuum Potts model with background interaction. This idea was then used in a variety of articles, for instance to prove phase transition for the symmetric Widom-Rowlinson model with unbounded radii [9,17]. The idea of the Fortuin-Kasteleyn representation is generalized to the non-symmetric case in the present article.…”