Phase transitions in Delaunay Potts models

TL;DR

This paper proves phase transitions in continuum Delaunay Potts models with infinite-range repulsive interactions, extending previous finite-range results and confirming a long-standing conjecture.

Contribution

It introduces a Delaunay random-cluster representation for geometry-dependent multi-body systems, advancing understanding of phase transitions in such models.

Findings

Established phase transitions for Delaunay Potts models with infinite-range interactions.

Extended finite-range results to infinite-range decay interactions.

Confirmed a conjecture of Lebowitz and Lieb regarding phase transitions.

Abstract

We establish phase transitions for classes of continuum Delaunay multi-type particle systems (continuum Potts models) with infinite range repulsive interaction between particles of different type. In one class of the Delaunay Potts models studied the repulsive interaction is a triangle (multi-body) interaction whereas in the second class the interaction is between pairs (edges) of the Delaunay graph. The result for the edge model is an extension of finite range results in \cite{BBD04} for the Delaunay graph and in \cite{GH96} for continuum Potts models to an infinite range repulsion decaying with the edge length. This is a proof of an old conjecture of Lebowitz and Lieb. The repulsive triangle interactions have infinite range as well and depend on the underlying geometry and thus are a first step towards studying phase transitions for geometry-dependent multi-body systems. Our approach involves a Delaunay random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts model. The phase transitions manifest themselves in the percolation of the corresponding random-cluster model. Our proofs rely on recent studies \cite{DDG12} of Gibbs measures for geometry-dependent interactions.