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

Vincent Beffara (1) , H Duminil-Copin (2) , Stanislas Smirnov (3)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 Section de mathématiques [Genève]
3 Chebyshev Laboratory

The critical surface for random-cluster model with cluster-weight q ≥ 4 on iso-radial 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 [6] to a large class of planar graphs. In particular, the anisotropic random-cluster model on the square lattice is shown to be critical if pvp h (1−pv)(1−p h) = q, where p v and p h denote the horizontal and vertical edge-weights respectively. We also provide consequences for Potts models.

Domaines: Mathématiques [math]

Vincent Beffara, H Duminil-Copin, Stanislas Smirnov. On the critical parameters of the q ≥ 4 random-cluster model on isoradial graphs. Journal of Physics A: Mathematical and Theoretical, 2015, 48, pp.484003. ⟨10.1088/1751-8113/48/48/484003⟩. ⟨hal-01618594⟩