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Explicit isoperimetric constants and phase transitions in the random-cluster model. (English) Zbl 1025.60044

This paper investigates the “random cluster model” on non-amenable graphs. The random cluster model is a model of dependent percolation that is closely related to Ising and Potts models of statistical mechanics. The global dependence arises from weights that depend on the number of connected components of the set of occupied vertices. The paper proves relations between various critical values of the percolation density (related to different boundary conditions). The proofs involve explicit calculations of isoperimetric constants. A main result is the proof of absence of percolation for the free random cluster measure on all non-amenable quasi-transitive unimodular graphs at the lower critical point. Applications to Potts models on non-amenable graphs show the existence of phase transitions for entire intervals of temperatures.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
82B43 Percolation
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