Probability Theory and Related Fields

, Volume 153, Issue 3–4, pp 511–542 | Cite as

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

  • Vincent BeffaraEmail author
  • Hugo Duminil-Copin


We prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with parameter q ≥ 1 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 ≥ 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 ≥ 1, in contrast to earlier methods valid only for certain given q. The proof extends to the triangular and the hexagonal lattices as well.

Mathematics Subject Classification (2000)

60K35 82B20 (primary) 82B26 82B43 


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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Unité de Mathématiques Pures et AppliquéesÉcole Normale Supérieure de LyonLyon Cedex 7France
  2. 2.Département de MathématiquesUniversité de GenèveGenevaSwitzerland

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