Phase Transitions in Delaunay Potts Models

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• 作者：Stefan Adams ; Michael Eyers
• 关键词：Delaunay tessellation ; Gibbs measures ; Random cluster measures ; Percolation ; Phase transitions ; Coarse graining ; Multi ; body interaction
• 刊名：Journal of Statistical Physics
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：162
• 期：1
• 页码：162-185
• 全文大小：822 KB
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• 作者单位：Stefan Adams (1)
  Michael Eyers (1)
  
  1. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Statistical Physics
  Mathematical and Computational Physics
  Physical Chemistry
  Quantum Physics
  
• 出版者：Springer Netherlands
• ISSN：1572-9613

文摘

We establish phase transitions for certain 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 Bertin et al. (J Stat Phys 114(1–2):79–100, 2004) for the Delaunay graph and in Georgii and Häggström (Commun Math Phys 181:507–528, 1996) 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 (Dereudre et al. in Probab Theory Relat Fields 153:643–670, 2012) of Gibbs measures for geometry-dependent interactions.