The Annals of Probability

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

Abstract

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

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

Dates
Revised: July 2018
First available in Project Euclid: 4 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1562205701

Digital Object Identifier
doi:10.1214/18-AOP1301

Mathematical Reviews number (MathSciNet)
MR3980915

Zentralblatt MATH identifier
07114711

Citation

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. https://projecteuclid.org/euclid.aop/1562205701

References

• [1] Aru, J., Sepulveda, A. and Werner, W. On bounded-type thin local sets of the two-dimensional Gaussian free field. Available at arXiv:1603.0336v2.
• [2] Beffara, V. (2008). The dimension of the SLE curves. Ann. Probab. 36 1421–1452.
• [3] Beffara, V. and Duminil-Copin, H. (2012). The self-dual point of the two-dimensional random-cluster model is critical for $q\geq 1$. Probab. Theory Related Fields 153 511–542.
• [4] Benoist, S., Duminil-Copin, H. and Hongler, C. (2016). Conformal invariance of crossing probabilities for the Ising model with free boundary conditions. Ann. Inst. Henri Poincaré Probab. Stat. 52 1784–1798.
• [5] Camia, F. and Newman, C. M. (2006). Two-dimensional critical percolation: The full scaling limit. Comm. Math. Phys. 268 1–38.
• [6] Camia, F. and Newman, C. M. (2007). Critical percolation exploration path and $\mathrm{SLE}_{6}$: A proof of convergence. Probab. Theory Related Fields 139 473–519.
• [7] Chelkak, D., Duminil-Copin, H. and Hongler, C. (2016). Crossing probabilities in topological rectangles for the critical planar FK-Ising model. Electron. J. Probab. 21 Paper No. 5, 28.
• [8] Chelkak, D., Duminil-Copin, H., Hongler, C., Kemppainen, A. and Smirnov, S. (2014). Convergence of Ising interfaces to Schramm’s SLE curves. C. R. Math. Acad. Sci. Paris 352 157–161.
• [9] Chelkak, D., Hongler, C. and Izyurov, K. (2015). Conformal invariance of spin correlations in the planar Ising model. Ann. of Math. (2) 181 1087–1138.
• [10] Chelkak, D. and Smirnov, S. (2012). Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189 515–580.
• [11] Friedli, S. and Velenik, Y. (2018). Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction. Cambridge Univ. Press, Cambridge.
• [12] Grimmett, G. (2006). The Random-Cluster Model. Grundlehren der Mathematischen Wissenschaften 333. Springer, Berlin.
• [13] Hongler, C. (2010). Conformal Invariance of Ising Model Correlations. Ph.D. thesis, Univ. de Genève. Available at https://archive-ouverte.unige.ch/unige:18163.
• [14] Hongler, C. and Kytölä, K. (2013). Ising interfaces and free boundary conditions. J. Amer. Math. Soc. 26 1107–1189.
• [15] Hongler, C., Kytölä, K. and Viklund, F. J. Conformal field theory at the lattice level: Discrete complex analysis and Virasoro structures. Available at arXiv:1307.4104v2.
• [16] Hongler, C. and Smirnov, S. (2013). The energy density in the planar Ising model. Acta Math. 211 191–225.
• [17] Ising, E. (1925). Beitrag zur Theorie des Ferromagnetismus. Z. Phys. 31 253–258.
• [18] Izyurov, K. (2015). Smirnov’s observable for free boundary conditions, interfaces and crossing probabilities. Comm. Math. Phys. 337 225–252.
• [19] Kemppainen, A. and Smirnov, S. Conformal invariance of boundary touching loops of FK Ising model. Available at arXiv:1509.08858.
• [20] Kemppainen, A. and Smirnov, S. Conformal invariance in random cluster models. II. Full scaling limit as a branching SLE. Available at arXiv:1609.08527.
• [21] Kemppainen, A. and Smirnov, S. (2017). Random curves, scaling limits and Loewner evolutions. Ann. Probab. 45 698–779.
• [22] Lawler, G. F. (2005). Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs 114. Amer. Math. Soc., Providence, RI.
• [23] Lawler, G. F., Schramm, O. and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 939–995.
• [24] Miller, J. and Sheffield, S. CLE(4) and the Gaussian free field. Preprint.
• [25] Miller, J., Sheffield, S. and Werner, W. (2017). CLE percolations. Forum Math. Pi 5 e4, 102.
• [26] Pfister, C.-E. and Velenik, Y. (1999). Interface, surface tension and reentrant pinning transition in the $2$D Ising model. Comm. Math. Phys. 204 269–312.
• [27] Pommerenke, C. (1992). Boundary Behaviour of Conformal Maps. Grundlehren der Mathematischen Wissenschaften 299. Springer, Berlin.
• [28] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221–288.
• [29] Schramm, O. and Sheffield, S. (2009). Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202 21–137.
• [30] Schramm, O. and Wilson, D. B. (2005). SLE coordinate changes. New York J. Math. 11 659–669.
• [31] Sheffield, S. (2009). Exploration trees and conformal loop ensembles. Duke Math. J. 147 79–129.
• [32] Sheffield, S. and Werner, W. (2012). Conformal loop ensembles: The Markovian characterization and the loop-soup construction. Ann. of Math. (2) 176 1827–1917.
• [33] Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 239–244.
• [34] Smirnov, S. (2006). Towards conformal invariance of 2D lattice models. In International Congress of Mathematicians. Vol. II 1421–1451. Eur. Math. Soc., Zürich.
• [35] Smirnov, S. (2010). Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math. (2) 172 1435–1467.