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{1020107775,
author = {H\"aggstr\"om, Olle and Jonasson, Johan and Lyons, Russell},
title = {Explicit Isoperimetric Constants and Phase Transitions in the
Random-Cluster Model},
journal = {Ann. Probab.},
volume = {30},
number = {1},
year = {2002},
pages = { 443-473},
language = {en},
url = {http://dml.mathdoc.fr/item/1020107775}
}
Häggström, Olle; Jonasson, Johan; Lyons, Russell. Explicit Isoperimetric Constants and Phase Transitions in the
Random-Cluster Model. Ann. Probab., Tome 30 (2002) no. 1, pp. 443-473. http://gdmltest.u-ga.fr/item/1020107775/