The Annals of Probability

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

Olle Häggström, Johan Jonasson, and Russell Lyons

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

First available in Project Euclid: 29 April 2002

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B26: Phase transitions (general) 82B43: Percolation [See also 60K35]

Percolation Ising model Potts medel planar graph planar dual nonamenable graph robust phase transition


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.

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