Open Access
2019 The Widom-Rowlinson model on the Delaunay graph
Stefan Adams, Michael Eyers
Electron. J. Probab. 24: 1-41 (2019). DOI: 10.1214/19-EJP370


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.


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Stefan Adams. Michael Eyers. "The Widom-Rowlinson model on the Delaunay graph." Electron. J. Probab. 24 1 - 41, 2019.


Received: 24 May 2018; Accepted: 4 October 2019; Published: 2019
First available in Project Euclid: 11 October 2019

zbMATH: 07142908
MathSciNet: MR4029417
Digital Object Identifier: 10.1214/19-EJP370

Primary: 60G55 , 60G57 , 82B05 , 82B21 , 82B26 , 82B43

Keywords: coarse graining , Delaunay tessellation , Gibbs measures , mixed site-bond percolation , multi-body interaction , phase transition , random cluster measures , Widom-Rowlinson

Vol.24 • 2019
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