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

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

2000-08-24

arxiv.org/abs/math/0008191v2

Ann. Probab. 30 (2002), 443--473

Mathematics

Probability

Scientific paper

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 $\pc^\f (q) > \pu^\w (q)$ 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.

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