The self-dual point of the two-dimensional random-cluster model is critical for $q\geq 1$

Vincent Beffara; Hugo Duminil-Copin

doi:10.1007/s00440-011-0353-8

The self-dual point of the two-dimensional random-cluster model is critical for $q\geq 1$

Vincent Beffara ¹ Hugo Duminil-Copin ²

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

Abstract : We prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with parameter $q\geq1$ on the square lattice is equal to the self-dual point $p_{sd}(q) = \sqrt q /(1+\sqrt q)$. This gives a proof that the critical temperature of the $q$-state Potts model is equal to $\log (1+\sqrt q)$ for all $q\geq 2$. We further prove that the transition is sharp, meaning that there is exponential decay of correlations in the sub-critical phase. The techniques of this paper are rigorous and valid for all $q\geq 1$, in contrast to earlier methods valid only for certain given $q$. The proof extends to the triangular and the hexagonal lattices as well.

Keywords : Percolation Random-Cluster Model Critical Point

Type de document :

Article dans une revue

Probability Theory and Related Fields, Springer Verlag, 2012, 153 (3-4), pp.511 - 542. 〈10.1007/s00440-011-0353-8〉

Domaine :

Mathématiques [math] / Probabilités [math.PR]

Physique [physics] / Physique mathématique [math-ph]

Mathématiques [math] / Physique mathématique [math-ph]

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https://hal-ens-lyon.archives-ouvertes.fr/ensl-00495872
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