Abstract
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.
Highlights
Introduction and results1.1 IntroductionThis paper is a continuation of the study in [AE16] concerning phase transition in continuum systems
The focus of this research is on another specific model, the Widom and Rowlinson model [WR70], for which a phase transition is known to occur, and on a novel counting technique for connected components of the corresponding Delaunay random cluster model
The main novelty of our paper is a uniform bound on the number of connected components in the Delaunay random cluster model which is purely based on geometrical properties of Delaunay tessellations in two dimensions
Summary
This paper is a continuation of the study in [AE16] concerning phase transition in continuum systems. In this paper we establish the existence of a phase transition for a class of continuum Delaunay Widom-Rowlinson (Potts) models in R2. The repulsive interaction between unlike particles is of finite range, and it depends on the geometry of the Delaunay tessellation, i.e., the length of the edges. There are two novelties in this research: We obtain a phase transition for soft repulsion (no hard-core repulsion) between unlike particles on the Delaunay structure with no additional constraints on the distribution of the underlying particle system. This paper is an extensive further development of the recent work [AE16] where all models had an additional background hard-core potential introducing a length scale for the configurations. Our results are extension of [LL72, Rue71] and [CCK95] to the Delaunay structure replacing hard-core constraint by our soft-core repulsion. In the non-symmetric Widom-Rowlinson models, the existence of a phase transition has been established by Bricmont et al [BKL84], and recently by Suhov et al [MSS15]
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