Open Access
Translator Disclaimer
July 2019 The scaling limit of critical Ising interfaces is $\mathrm{CLE}_{3}$
Stéphane Benoist, Clément Hongler
Ann. Probab. 47(4): 2049-2086 (July 2019). DOI: 10.1214/18-AOP1301


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].


Download Citation

Stéphane Benoist. Clément Hongler. "The scaling limit of critical Ising interfaces is $\mathrm{CLE}_{3}$." Ann. Probab. 47 (4) 2049 - 2086, July 2019.


Received: 1 May 2016; Revised: 1 July 2018; Published: July 2019
First available in Project Euclid: 4 July 2019

zbMATH: 07114711
MathSciNet: MR3980915
Digital Object Identifier: 10.1214/18-AOP1301

Primary: 60J67 , 60K35 , 82B20 , 82B27

Keywords: conformal invariance , Conformal loop ensembles , Criticality , Duality , Fortuin–Kasteleyn random-cluster model , free boundary conditions , Ising model , phase transition , random curves , scaling limits , Schramm–Loewner evolution

Rights: Copyright © 2019 Institute of Mathematical Statistics


Vol.47 • No. 4 • July 2019
Back to Top