Explicit isoperimetric constants and phase transitions in the random-cluster model

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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≥ 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 (q) > (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 non-robust phase transition for the Potts model on nonamenable regular graphs.