Phase transition for Gibbs Delaunay Tessellations with geometric hardcore conditions

TL;DR

This paper proves the existence of multiple Gibbs Delaunay Potts tessellations with hardcore conditions, showing phase transitions for large activity and interaction strength using coarse-graining and percolation comparison.

Contribution

It establishes the existence of multiple translation-invariant Gibbs Delaunay tessellations under hardcore constraints, a novel result in geometric particle systems.

Findings

Existence of multiple Gibbs Delaunay tessellations for large parameters

Phase transition characterized by multiple invariant states

Application of coarse-graining and percolation techniques

Abstract

In this paper, we prove the existence of infinite Gibbs Delaunay Potts tessellations for marked particle configurations. The particle systems has two types of interaction, a so-called \emph{background potential} ensures that small and large triangles are excluded in the Delaunay tessellation, and is similar to the so-called hardcore potential introduced in \cite{Der08}. Particles carry one of $q$ separate marks. Our main result is that for large activities and high \emph{type interaction} strength the model has at least $ q$ distinct translation-invariant Gibbs Delaunay Potts tessellations. The main technique is a coarse-graining procedure using the scales in the system followed by comparison with site percolation on $ \Z^2 $.