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The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1
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  • Published: 18 March 2011

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

  • Vincent Beffara1 &
  • Hugo Duminil-Copin2 

Probability Theory and Related Fields volume 153, pages 511–542 (2012)Cite this article

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Abstract

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.

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Authors and Affiliations

  1. Unité de Mathématiques Pures et Appliquées, École Normale Supérieure de Lyon, 69364, Lyon Cedex 7, France

    Vincent Beffara

  2. Département de Mathématiques, Université de Genève, Geneva, Switzerland

    Hugo Duminil-Copin

Authors
  1. Vincent Beffara
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  2. Hugo Duminil-Copin
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Correspondence to Vincent Beffara.

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Beffara, V., Duminil-Copin, H. The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1. Probab. Theory Relat. Fields 153, 511–542 (2012). https://doi.org/10.1007/s00440-011-0353-8

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  • Received: 29 November 2010

  • Revised: 21 February 2011

  • Published: 18 March 2011

  • Issue Date: August 2012

  • DOI: https://doi.org/10.1007/s00440-011-0353-8

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Mathematics Subject Classification (2000)

  • 60K35
  • 82B20 (primary)
  • 82B26
  • 82B43
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