# The Widom-Rowlinson Model on the Delaunay Graph

## TL;DR

This paper proves phase transitions in continuum Delaunay multi-type particle systems with edge-length-dependent interactions, extending the Lebowitz-Lieb conjecture and using geometric and random-cluster representations.

## Contribution

It establishes phase transitions for Delaunay Widom-Rowlinson models with geometric interactions, introducing a novel geometric approach and a Delaunay random-cluster representation.

## Key findings

- Phase transition occurs at large activities and potential parameters.
- A Delaunay random-cluster model is used to analyze percolation.
- Shorter Delaunay edges are more likely to be open, controlling component size.

## Abstract

We establish phase transitions for continuum Delaunay multi-type particle systems (continuum Potts or Widom-Rowlinson models) with a repulsive interaction between particles of different types. Our interaction potential depends solely on the length of the Delaunay edges. We show that a phase transition occurs for sufficiently large activities and for sufficiently large potential parameter proving 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 Delaunay tessellations in $\mathbb{R}^2 $ and on recent studies [DDG12] 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. The interaction potential ensures that shorter Delaunay edges are more likely to be open and thus offsets the possibility of having an unbounded number of connected components.