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The Near-Critical Planar FK-Ising Model

Communications in Mathematical Physics volume 326, pages 1–35 (2014)Cite this article

Abstract

We study the near-critical FK-Ising model. First, a determination of the correlation length defined via crossing probabilities is provided. Second, a phenomenon about the near-critical behavior of the FK-Ising is highlighted, which is completely missing from the case of standard percolation: in any monotone coupling of FK configurations ω p (e.g., in the one introduced in Grimmett (Ann Probab 23(4):1461–1510, 1995)), as one raises p near p c , the new edges arrive in a self-organized way, so that the correlation length is not governed anymore by the number of pivotal edges at criticality.

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Authors and Affiliations

  1. Département de Mathématiques, Université de Genève, 2-4 rue du lièvre, Case postate 64, 1211, Geneva 4, Switzerland

    Hugo Duminil-Copin

  2. CNRS & Ecole Normale Supérieure de Lyon, UMPA 46, allée d’Italie, 69364, Lyon Cedex 07, France

    Christophe Garban

  3. Institute of Mathematics, Technical University of Budapest, Egry József u. 1, 1111, Budapest, Hungary

    Gábor Pete

  4. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    Gábor Pete

Authors
  1. Hugo Duminil-Copin
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  2. Christophe Garban
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  3. Gábor Pete
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Corresponding author

Correspondence to Christophe Garban.

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Communicated by M. Aizenman

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Duminil-Copin, H., Garban, C. & Pete, G. The Near-Critical Planar FK-Ising Model. Commun. Math. Phys. 326, 1–35 (2014). https://doi.org/10.1007/s00220-013-1857-0

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  • DOI: https://doi.org/10.1007/s00220-013-1857-0

Keywords

  • Correlation Length
  • Conformal Invariance
  • Free Boundary Condition
  • Dual Cluster
  • Fermionic Observable