Phase Transitions in Continuum Delaunay Potts and Widom-Rowlinson 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 an FK (Fortuin-Kasteleyn)  random cluster representation adapted to the Delaunay structure and percolation in the FK model via coarse graining methods.  In a second part we present our results for the Delaunay Widom-Rowlinson model in dimension two with an repulsion depending smoothly on the length of the Delaunay edges. This one is an extension of the work by Chayes et al on the Widom-Rowlinson model to the Delaunay graph and to replace hard-core repulsion by soft repulsion. In addition our approach does not require a hard core background potential as previous studies. The  soft repulsion requires novel purely geometrical arguments which we briefly outline. (Joint work with E. Eyers)