Phase Transitions in Continuum Delaunay Potts Models

We discuss recent results on phase transitions of Delaunay Potts models in dimension two where the interaction depends on Delaunay edges respectively Delaunay triangles.
This work is an extension of the Lebowitz & Lieb soft-core continuum Potts model to geometrically dependent interaction systems. The main tool is a FK (Fortuin-Kasteleyn) random cluster representation adapted to the Delaunay structure and percolation in the FK model. If time permits we discuss the Voronoi-Ising model where the interaction is function of the length of the common boundary and its connection to variants of the RSW (Russo-Seymour-Welsh) theorem in continuum percolation.