| Explicit Isoperimetric Constants and Phase Transitions in the Random-Cluster Model (2001) | |||||||||||||||||
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 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 u (q) for q large enough. The latter follows from considerations of isoperimetric constants, and we give the rst nontrivial explicit calculations of such constants. Such considerations are also used to prove non-robust phase transition for the Potts model on nonamenable regular graphs. | |||||||||||||||||
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