Percolation results for the continuum random cluster model. (20th March 2018)

• Record Type: Journal Article
• Authors: Houdebert, Pierre
• Is Part Of: Advances in applied probability. Volume 50:Number 1(2018)
• Page Start: 231
• Page End: 244
• Publication Date: 2018-03-20
• Subjects: Gibbs point process, -- Widom–Rowlinson model, -- stochastic comparison, -- Fortuin–Kasteleyn representation; Primary 60D05, -- Secondary 60G10, -- 60G55, -- 60G57, -- 60G60, -- 60K35, -- 82B21, -- 82B26, -- 82B43; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2018.11
• Languages: English
• ISSNs: 0001-8678

Abstract

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.