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The Near-Critical Planar FK-Ising Model

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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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