  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.
