The Annals of Probability

The scaling limit of critical Ising interfaces is $\mathrm{CLE}_{3}$

Stéphane Benoist and Clément Hongler

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In this paper, we consider the set of interfaces between $+$ and $-$ spins arising for the critical planar Ising model on a domain with $+$ boundary conditions, and show that it converges to nested CLE$_{3}$.

Our proof relies on the study of the coupling between the Ising model and its random cluster (FK) representation, and of the interactions between FK and Ising interfaces. The main idea is to construct an exploration process starting from the boundary of the domain, to discover the Ising loops and to establish its convergence to a conformally invariant limit. The challenge is that Ising loops do not touch the boundary; we use the fact that FK loops touch the boundary (and hence can be explored from the boundary) and that Ising loops in turn touch FK loops, to construct a recursive exploration process that visits all the macroscopic loops.

A key ingredient in the proof is the convergence of Ising free arcs to the Free Arc Ensemble (FAE), established in [Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016) 1784–1798]. Qualitative estimates about the Ising interfaces then allow one to identify the scaling limit of Ising loops as a conformally invariant collection of simple, disjoint $\mathrm{SLE}_{3}$-like loops, and thus by the Markovian characterization of Sheffield and Werner [Ann. of Math. (2) 176 (2012) 1827–1917] as a $\mathrm{CLE}_{3}$.

A technical point of independent interest contained in this paper is an investigation of double points of interfaces in the scaling limit of critical FK-Ising. It relies on the technology of Kemppainen and Smirnov [Ann. Probab. 45 (2017) 698–779].

Article information

Ann. Probab., Volume 47, Number 4 (2019), 2049-2086.

Received: May 2016
Revised: July 2018
First available in Project Euclid: 4 July 2019

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Zentralblatt MATH identifier

Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE) 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B27: Critical phenomena 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Ising model phase transition free boundary conditions Fortuin–Kasteleyn random-cluster model criticality duality scaling limits conformal invariance random curves Schramm–Loewner evolution conformal loop ensembles


Benoist, Stéphane; Hongler, Clément. The scaling limit of critical Ising interfaces is $\mathrm{CLE}_{3}$. Ann. Probab. 47 (2019), no. 4, 2049--2086. doi:10.1214/18-AOP1301.

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