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January 2002 Explicit Isoperimetric Constants and Phase Transitions in the Random-Cluster Model
Olle Häggström, Johan Jonasson, Russell Lyons
Ann. Probab. 30(1): 443-473 (January 2002). DOI: 10.1214/aop/1020107775


The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results are obtained for the random-cluster model on nonamenable graphs with cluster parameter $q \geq 1$. Among these, the main ones are the absence of percolation for the free random-cluster measure at the critical value and examples of planar regular graphs with regular dual where $p_ \mathrm{c}^{\mathrm{free}} (q) > p_ \mathrm{u}^{\mathrm{wired}} (q)$ for $q$ large enough. The latter follows from considerations of isoperimetric constants, and we give the first nontrivial explicit calculations of such constants. Such considerations are also used to prove nonrobust phase transition for the Potts model on nonamenable regular graphs.


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Olle Häggström. Johan Jonasson. Russell Lyons. "Explicit Isoperimetric Constants and Phase Transitions in the Random-Cluster Model." Ann. Probab. 30 (1) 443 - 473, January 2002.


Published: January 2002
First available in Project Euclid: 29 April 2002

zbMATH: 1025.60044
Digital Object Identifier: 10.1214/aop/1020107775

Primary: 60K35 , 82B20 , 82B26 , 82B43

Keywords: Ising model , nonamenable graph , percolation , planar graph planar dual , Potts medel , robust phase transition

Rights: Copyright © 2002 Institute of Mathematical Statistics


Vol.30 • No. 1 • January 2002
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