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 FK-Ising is highlighted, which is completely missing from the case of standard percolation: in any monotone coupling of FK configurations ${\omega }_p$ (e.g., in the one introduced in [Gri95]), 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.

34 pages, 8 figures. This is a streamlined version; the previous one contains more explanations and additional material on exceptional times in FK models with general $q$. Furthermore, the statement and proof of Theorem 1.2 have slightly changed

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• The near-critical planar FK-Ising model
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Cited by in corpus (13)

• Continuity of the phase transition for planar random-cluster and Potts models with $1\le q\le 4$
• Revisiting the combinatorics of the 2D Ising model
• The near-critical planar FK-Ising model
• Percolation since Saint-Flour
• Crossing probabilities in topological rectangles for the critical planar FK-Ising model
• Massive Scaling Limit of the Ising Model: Subcritical Analysis and Isomonodromy
• Convergence of Fermionic Observables in the Massive Planar FK-Ising Model
• Local time on the exceptional set of dynamical percolation, and the Incipient Infinite Cluster
• Some toy models of self-organized criticality in percolation
• A planar Ising model of self-organized criticality
• A Russo Seymour Welsh Theorem for critical site percolation on \(\mathbb{Z}^2\)
• Limit of the Wulff Crystal when approaching criticality for site percolation on the triangular lattice
• On the construction of discrete fermions in the FK-Ising model