On the critical parameters of the q≥4 random-cluster model on isoradial graphs

Monday, 6 July, 2015

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The critical surface for random-cluster model with cluster-weight q≥4 on isoradial graphs is identified using parafermionic observables. Correlations are also shown to decay exponentially fast in the subcritical regime. While this result is restricted to random-cluster models with q≥4, it extends the recent theorem of the two first authors to a large class of planar graphs. In particular, the anisotropic random-cluster model on the square lattice is shown to be critical if pvph / (1−pv)(1−ph) = q, where pv and ph denote the horizontal and vertical edge-weights respectively. We also mention consequences for Potts models.


Vincent Beffara
Hugo Duminil-Copin
Stanislav Smirnov