Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Conformal invariance of crossing probabilities for the Ising model with free boundary conditions

Stéphane Benoist, Hugo Duminil-Copin, and Clément Hongler

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We prove that crossing probabilities for the critical planar Ising model with free boundary conditions are conformally invariant in the scaling limit, a phenomenon first investigated numerically by Langlands, Lewis and Saint-Aubin (J. Stat. Phys. 98 (2000) 131–244). We do so by establishing the convergence of certain exploration processes towards $\operatorname{SLE}(3,\frac{-3}{2},\frac{-3}{2})$. We also construct an exploration tree for free boundary conditions, analogous to the one introduced by Sheffield (Duke Math. J. 147 (2009) 79–129).


Nous prouvons que les probabilités de croisement pour le modèle d’Ising planaire critique avec conditions aux bords libres sont invariantes conformes à la limite d’échelle, un phénomène initialement étudié numériquement par Langlands, Lewis et Saint-Aubin (J. Stat. Phys. 98 (2000) 131–244). Pour ce faire, nous établissons la convergence de certains processus d’exploration vers $\operatorname{SLE}(3,\frac{-3}{2},\frac{-3}{2})$. Nous construisons également un arbre d’exploration pour les conditions aux bords libres, similaire à l’arbre d’exploration introduit par Sheffield (Duke Math. J. 147 (2009) 79–129).

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 4 (2016), 1784-1798.

Received: 19 January 2015
Revised: 21 June 2015
Accepted: 7 July 2015
First available in Project Euclid: 17 November 2016

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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

Ising model Interfaces Schramm–Loewner evolution Phase transition Crossing probabilities Exploration trees


Benoist, Stéphane; Duminil-Copin, Hugo; Hongler, Clément. Conformal invariance of crossing probabilities for the Ising model with free boundary conditions. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 4, 1784--1798. doi:10.1214/15-AIHP698.

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