The Widom-Rowlinson Model on the Delaunay Graph

We establish phase transitions for continuum Delaunay multi-type particle systems (continuum Potts or Widom-Rowlinson models) with infinite range repulsive interaction between particles of different type. Our interaction potential depends solely on the length of the Delaunay edges and is scale invariant up to a parameter replacing the role of inverse temperature. In fact we show that a phase transition occurs for all activities for sufficiently large potential parameter confirming an old conjecture that if phase transition occurs on the Delaunay graph it will be independent of the activity. This is a proof of an old conjecture of Lebowitz and Lieb extended to the Delaunay structure. Our approach involves a Delaunay random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts model. The phase transition manifests itself in the mixed site-bond percolation of the corresponding random-cluster model. Our proofs rely mainly on geometric properties of the Delaunay tessellations in $\R^2 $ and on recent studies by Dereudre et al. of Gibbs measures for geometry-dependent interactions. The main tool is a uniform bound on the number of connected components in the Delaunay graph which provides a novel approach to Delaunay Widom Rowlinson models based on purely geometric arguments.