On Delaunay random cluster models

Abstract

We examine continuum percolative problems on the Delaunay hypergraph structure. In particular, we investigate the existence of a percolation transition for a class of Gibbsian particle systems with random hyperedges between groups of particles. Each such system will take the form of a random cluster representation of a corresponding continuum Potts model with geometric interactions on hyperedges of the Delaunay hypergraph structure. Any percolation results in the random cluster representation will lead to the existence of a phase transition for the continuum Potts model: that is, the existence of more than one Gibbs measure. The original components of this research are as follows. After extending the random cluster representation of [GH96] to hypergraph structures, we achieve a phase transition for Delaunay continuum Potts models with infinite range type interactions – extending the work of [BBD03] in the process. Our main result is the existence of a phase transition for Delaunay continuum Potts models with no background interaction and just a soft type interaction. This is an extension of the phase transition results for the hardcore (resp. softcore) Widom–Rowlinson model of [R71] and later [CCK94], (resp. [LL72]). Our final piece of originality comes in the guise of an overview of the obstacles faced when investigating further percolative problems in the Delaunay hypergraph structure such as the Russo–Seymour–Welsh Theorem