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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Abstract

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).

Résumé

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1479373248

Digital Object Identifier
doi:10.1214/15-AIHP698

Mathematical Reviews number (MathSciNet)
MR3573295

Zentralblatt MATH identifier
1355.60119

Subjects
Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE) 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

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

Citation

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. https://projecteuclid.org/euclid.aihp/1479373248


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