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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References

  • [1] R. J. Baxter. Exactly Solved Models in Statistical Mechanics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1989. Reprint of the 1982 original.
  • [2] J. L. Cardy. Critical percolation in finite geometries. J. Phys. A 25 (4) (1992) L201–L206.
  • [3] D. Chelkak and S. Smirnov. Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. math. (2012) 189: 515.
  • [4] D. Chelkak, H. Duminil-Copin and C. Hongler. Crossing probabilities in topological rectangles for the critical planar FK-Ising model. Electronic J. Prob., 21 (2016), paper no. 5, 28 pp.
  • [5] D. Chelkak, H. Duminil-Copin, C. Hongler, A. Kemppainen and S. Smirnov. Convergence of Ising interfaces to Schramm’s SLE curves. C. R. Math. Acad. Sci. Paris 352 (2) (2014) 157–161.
  • [6] D. Chelkak, C. Hongler and K. Izyurov. Conformal invariance of spin correlations in the planar Ising model. Ann. of Math. (2) 181 (3) (2015) 1087–1138.
  • [7] D. Chelkak and S. Smirnov. Discrete complex analysis on isoradial graphs. Adv. Math. 228 (3) (2011) 1590–1630.
  • [8] P. Di Francesco, P. Mathieu and D. Sénéchal. Conformal Field Theory, Corrected edition. Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 1999.
  • [9] M. E. Fisher. Renormalization group theory: Its basis and formulation in statistical physics. In Conceptual Foundations of Quantum Field Theory (Boston, MA, 1996) 89–135. Cambridge Univ. Press, Cambridge, 1999.
  • [10] C. Hongler and K. Kytölä. Ising interfaces and free boundary conditions. J. Amer. Math. Soc. 26 (4) (2013) 1107–1189.
  • [11] K. Izyurov. Holomorphic spinor observables and interfaces in the critical Ising model. Ph.D. thesis, Université de Genève, 2011. Available at http://archive-ouverte.unige.ch/unige:18424.
  • [12] K. Izyurov. Smirnov’s observable for free boundary conditions, interfaces and crossing probabilities. Comm. Math. Phys. 337 (1) (2015) 225–252.
  • [13] A. Kemppainen and S. Smirnov. Random curves, scaling limits and Loewner evolutions. ArXiv e-prints, 2012.
  • [14] H. A. Kramers and G. H. Wannier. Statistics of the two-dimensional ferromagnet. I. Phys. Rev. (2) 60 (1941) 252–262.
  • [15] R. Langlands, P. Pouliot and Y. Saint-Aubin. Conformal invariance in two-dimensional percolation. Bull. Amer. Math. Soc. (N.S.) 30 (1) (1994) 1–61.
  • [16] R. P. Langlands, M. Lewis and Y. Saint-Aubin. Universality and conformal invariance for the Ising model in domains with boundary. J. Stat. Phys. 98 (1–2) (2000) 131–244.
  • [17] G. F. Lawler. Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs 114. American Mathematical Society, Providence, RI, 2005.
  • [18] B. McCoy and T. T. Wu. The Two-Dimensional Ising Model. Harvard Univ. Press, Boston, MA, 1973.
  • [19] J. Miller and S. Sheffield. Imaginary geometry I: Interacting SLEs. ArXiv e-prints, 2012. Prob. Theor. and Rel. Fields, to appear.
  • [20] L. Onsager. Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. (2) 65 (1944) 117–149.
  • [21] J. Palmer. Planar Ising Correlations. Progress in Mathematical Physics 49. Birkhäuser Boston Inc., Boston, MA, 2007.
  • [22] O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 (2000) 221–288.
  • [23] O. Schramm and D. B. Wilson. SLE coordinate changes. New York J. Math. 11 (2005) 659–669 (electronic).
  • [24] S. Sheffield. Exploration trees and conformal loop ensembles. Duke Math. J. 147 (1) (2009) 79–129.
  • [25] S. Smirnov. Towards conformal invariance of 2D lattice models. In International Congress of Mathematicians. Vol. II 1421–1451. Eur. Math. Soc., Zürich, 2006.