Electronic Journal of Probability

The Widom-Rowlinson model on the Delaunay graph

Stefan Adams and Michael Eyers

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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.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 114, 41 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes 60G57: Random measures 82B21: Continuum models (systems of particles, etc.) 82B05: Classical equilibrium statistical mechanics (general) 82B26: Phase transitions (general) 82B43: Percolation [See also 60K35]

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

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Adams, Stefan; Eyers, Michael. The Widom-Rowlinson model on the Delaunay graph. Electron. J. Probab. 24 (2019), paper no. 114, 41 pp. doi:10.1214/19-EJP370. https://projecteuclid.org/euclid.ejp/1570759239

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