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Phase transition in the random triangle model

Part of: Graph theory Equilibrium statistical mechanics

Published online by Cambridge University Press:  14 July 2016

Olle Häggström  and
Johan Jonasson
Olle Häggström*
Affiliation:
Chalmers University of Technology
Johan Jonasson*
Affiliation:
Chalmers University of Technology
*
∗Postal address: Department of Mathematics, Chalmers University of Technology and Göteborg University, S41296 Göteborg, Sweden.
∗Postal address: Department of Mathematics, Chalmers University of Technology and Göteborg University, S41296 Göteborg, Sweden.
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Abstract

The random triangle model was recently introduced as a random graph model that captures the property of transitivity that is often found in social networks, i.e. the property that given that two vertices are second neighbors, they are more likely to be neighbors. For parameters p ∊ [0,1] and q ≥ 1, and a finite graph G = (V, E), it assigns to elements η of {0,1}E probabilities which are proportional to where t(η) is the number of triangles in the open subgraph. In this paper the behavior of the random triangle model on the two-dimensional triangular lattice is studied. By mapping the system onto an Ising model with external field on the hexagonal lattice, it is shown that phase transition occurs if and only if p = (q−1)−2/3 and q > qc for a critical value qc which turns out to equal It is furthermore demonstrated that phase transition cannot occur unless p = pc(q), the critical value for percolation of open edges for given q. This implies that for q ≥ qc, pc(q) = (q−1)−2/3.

Keywords

Percolationplanar dualitystochastic dominationIsing model

MSC classification

Primary: 05C80: Random graphs 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1101 - 1115
Copyright
Copyright © Applied Probability Trust 1999 

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