## The Annals of Probability

### Explicit Isoperimetric Constants and Phase Transitions in the Random-Cluster Model

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 30, Number 1 (2002), 443-473.

Dates
First available in Project Euclid: 29 April 2002

https://projecteuclid.org/euclid.aop/1020107775

Digital Object Identifier
doi:10.1214/aop/1020107775

Mathematical Reviews number (MathSciNet)
MR1894115

Zentralblatt MATH identifier
1025.60044

#### Citation

Häggström, Olle; Jonasson, Johan; Lyons, Russell. Explicit Isoperimetric Constants and Phase Transitions in the Random-Cluster Model. Ann. Probab. 30 (2002), no. 1, 443--473. doi:10.1214/aop/1020107775. https://projecteuclid.org/euclid.aop/1020107775

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• BLOOMINGTON, INDIANA 47405 E-MAIL: rdlyons@indiana.edu