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Markov random fields and percolation on general graphs

Part of: Special processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  01 July 2016

Olle Häggström
Olle Häggström*
Affiliation:
Chalmers University of Technology
*
∗ Postal address: Department of Mathematics, Chalmers University of Technology, S-412 96 Göteborg, Sweden.
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Abstract

Let G be an infinite, locally finite, connected graph with bounded degree. We show that G supports phase transition in all or none of the following five models: bond percolation, site percolation, the Ising model, the Widom-Rowlinson model and the beach model. Some, but not all, of these implications hold without the bounded degree assumption. We finally give two examples of (random) unbounded degree graphs in which phase transition in all five models can be established: supercritical Galton-Watson trees, and Poisson-Voronoi tessellations of ℝd for d ≥ 2.

Keywords

PercolationIsing modelWidom-Rowlinson modelbeach modelGalton-Watson treePoisson-Voronoi tessellation

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B43: Percolation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 1 , March 2000 , pp. 39 - 66
Copyright
Copyright © Applied Probability Trust 2000 

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