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CLE PERCOLATIONS

  • JASON MILLER (a1), SCOTT SHEFFIELD (a2) and WENDELIN WERNER (a3)

Abstract

Conformal loop ensembles (CLEs) are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded by any loop is a canonical random connected fractal set — a random and conformally invariant analog of the Sierpinski carpet or gasket.

In the present paper, we derive a direct relationship between the CLEs with simple loops ( $\text{CLE}_{\unicode[STIX]{x1D705}}$ for $\unicode[STIX]{x1D705}\in (8/3,4)$ , whose loops are Schramm’s $\text{SLE}_{\unicode[STIX]{x1D705}}$ -type curves) and the corresponding CLEs with nonsimple loops ( $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ with $\unicode[STIX]{x1D705}^{\prime }:=16/\unicode[STIX]{x1D705}\in (4,6)$ , whose loops are $\text{SLE}_{\unicode[STIX]{x1D705}^{\prime }}$ -type curves). This correspondence is the continuum analog of the Edwards–Sokal coupling between the $q$ -state Potts model and the associated FK random cluster model, and its generalization to noninteger  $q$ .

Like its discrete analog, our continuum correspondence has two directions. First, we show that for each $\unicode[STIX]{x1D705}\in (8/3,4)$ , one can construct a variant of $\text{CLE}_{\unicode[STIX]{x1D705}}$ as follows: start with an instance of $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ , then use a biased coin to independently color each $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ loop in one of two colors, and then consider the outer boundaries of the clusters of loops of a given color. Second, we show how to interpret $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ loops as interfaces of a continuum analog of critical Bernoulli percolation within $\text{CLE}_{\unicode[STIX]{x1D705}}$ carpets — this is the first construction of continuum percolation on a fractal planar domain. It extends and generalizes the continuum percolation on open domains defined by $\text{SLE}_{6}$ and $\text{CLE}_{6}$ .

These constructions allow us to prove several conjectures made by the second author and provide new and perhaps surprising interpretations of the relationship between CLEs and the Gaussian free field. Along the way, we obtain new results about generalized $\text{SLE}_{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D70C})$ curves for $\unicode[STIX]{x1D70C}<-2$ , such as their decomposition into collections of $\text{SLE}_{\unicode[STIX]{x1D705}}$ -type ‘loops’ hanging off of $\text{SLE}_{\unicode[STIX]{x1D705}^{\prime }}$ -type ‘trunks’, and vice versa (exchanging $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D705}^{\prime }$ ). We also define a continuous family of natural $\text{CLE}$ variants called boundary conformal loop ensembles (BCLEs) that share some (but not all) of the conformal symmetries that characterize $\text{CLE}$ s, and that should be scaling limits of critical models with special boundary conditions. We extend the $\text{CLE}_{\unicode[STIX]{x1D705}}$ / $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ correspondence to a $\text{BCLE}_{\unicode[STIX]{x1D705}}$ / $\text{BCLE}_{\unicode[STIX]{x1D705}^{\prime }}$ correspondence that makes sense for the wider range $\unicode[STIX]{x1D705}\in (2,4]$ and $\unicode[STIX]{x1D705}^{\prime }\in [4,8)$ .

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Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

References

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[1] Aru, J., Sepúlveda, A. and Werner, W., ‘On bounded-type thin local sets of the two-dimensional Gaussian free field’, J. Inst. Math. Jussieu (2017), 1–28. doi:10.1017/S1474748017000160.
[2] Beffara, V., ‘The dimension of the SLE curves’, Ann. Probab. 36(4) (2008), 14211452.
[3] Camia, F., Garban, C. and Newman, C. M., ‘Planar Ising magnetization field I. Uniqueness of the critical scaling limit’, Ann. Probab. 43(2) (2015), 528571.
[4] Camia, F. and Newman, C. M., ‘Two-dimensional critical percolation: the full scaling limit’, Comm. Math. Phys. 268(1) (2006), 138.
[5] Cardy, J., ‘Conformal field theory and statistical mechanics’, inExact Methods in Low-dimensional Statistical Physics and Quantum Computing (Oxford University Press, Oxford, 2010), 6598.
[6] Chelkak, D., Duminil-Copin, H., Hongler, C., Kemppainen, A. and Smirnov, S., ‘Convergence of Ising interfaces to Schramm’s SLE curves’, C. R. Math. Acad. Sci. Paris 352(2) (2014), 157161.
[7] Chelkak, D., Hongler, C. and Izyurov, K., ‘Conformal invariance of spin correlations in the planar Ising model’, Ann. of Math. (2) 181(3) (2015), 10871138.
[8] Chelkak, D. and Smirnov, S., ‘Universality in the 2D Ising model and conformal invariance of fermionic observables’, Invent. Math. 189(3) (2012), 515580.
[9] Dubédat, J., ‘Commutation relations for Schramm–Loewner evolutions’, Comm. Pure Appl. Math. 60(12) (2007), 17921847.
[10] Dubédat, J., ‘Duality of Schramm-Loewner evolutions’, Ann. Sci. Éc. Norm. Supér. (4) 42(5) (2009), 697724.
[11] Dubédat, J., ‘SLE and the free field: partition functions and couplings’, J. Amer. Math. Soc. 22(4) (2009), 9951054.
[12] Duminil-Copin, H., Tassion, V. and Wu, H., 2017, in preparation.
[13] Duplantier, B., Miller, J. and Sheffield, S., ‘Liouville quantum gravity as a mating of trees’. ArXiv e-prints, 2014.
[14] Edwards, R. G. and Sokal, A. D., ‘Generalization of the Fortuin–Kasteleyn–Swendsen–Wang representation and Monte Carlo algorithm’, Phys. Rev. D (3) 38(6) (1988), 20092012.
[15] Fortuin, C. M. and Kasteleyn, P. W., ‘On the random-cluster model. I. Introduction and relation to other models’, Physica 57 (1972), 536564.
[16] Grimmett, G., The Random-Cluster Model, Grundlehren der Mathematischen Wissenschaften, 333 [Fundamental Principles of Mathematical Sciences] (Springer, Berlin, 2006).
[17] Gwynne, E., Mao, C. and Sun, X., ‘Scaling limits for the critical Fortuin–Kasteleyn model on a random planar map I: cone times’. ArXiv e-print, 2015.
[18] Gwynne, E. and Miller, J., ‘Convergence of the topology of critical Fortuin–Kasteleyn planar maps to that of CLE $_{\unicode[STIX]{x1D705}}$ on a Liouville quantum surface’, 2017, in preparation.
[19] Gwynne, E. and Sun, X., ‘Scaling limits for the critical Fortuin–Kastelyn model on a random planar map II: local estimates and empty reduced word exponent’. ArXiv e-print, 2015.
[20] Gwynne, E. and Sun, X., ‘Scaling limits for the critical Fortuin–Kastelyn model on a random planar map III: finite volume case’. ArXiv e-prints, 2015.
[21] Häggström, O., ‘Positive correlations in the fuzzy Potts model’, Ann. Appl. Probab. 9(4) (1999), 11491159.
[22] Higuchi, Y. and Wu, X.-Y., ‘Uniqueness of the critical probability for percolation in the two-dimensional Sierpiński carpet lattice’, Kobe J. Math. 25(1–2) (2008), 124.
[23] Hongler, C. and Kytölä, K., ‘Ising interfaces and free boundary conditions’, J. Amer. Math. Soc. 26(4) (2013), 11071189.
[24] Hongler, C. and Smirnov, S., ‘The energy density in the planar Ising model’, Acta Math. 211(2) (2013), 191225.
[25] Izyurov, K., ‘Smirnov’s observable for free boundary conditions, interfaces and crossing probabilities’, Comm. Math. Phys. 337(1) (2015), 225252.
[26] Kemppainen, A. and Smirnov, S., ‘Conformal invariance of boundary touching loops of FK Ising model’. ArXiv e-prints, 2015.
[27] Kemppainen, A. and S., Smirnov, ‘Random curves, scaling limits and Loewner evolutions’, Ann. Probab. 45(2) (2017), 698779.
[28] Kumagai, T., ‘Percolation on pre-Sierpinski carpets’, inNew Trends in Stochastic Analysis (Charingworth, 1994) (World Sci. Publ., River Edge, NJ, 1997), 288304.
[29] Lawler, G., Conformally Invariant Processes in the Plane, Mathematical Surveys and Monographs, 114 (American Mathematical Society, Providence, RI, 2005).
[30] Lawler, G., Schramm, O. and Werner, W., ‘Conformal restriction: the chordal case’, J. Amer. Math. Soc. 16(4) (2003), 917955. (electronic).
[31] Lawler, G., Schramm, O. and Werner, W., ‘Conformal invariance of planar loop-erased random walks and uniform spanning trees’, Ann. Probab. 32(1B) (2004), 939995.
[32] Maes, C. and Vande Velde, K., ‘The fuzzy Potts model’, J. Phys. A 28(15) (1995), 42614270.
[33] Miller, J. and Sheffield, S., ‘CLE(4) and the Gaussian free field’, 2017, in preparation.
[34] Miller, J. and Sheffield, S., ‘Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees’, Probab. Theory Related Fields (2017), doi:10.1007/s00440-017-0780-2.
[35] Miller, J. and Sheffield, S., ‘Liouville quantum gravity spheres as matings of finite-diameter trees’. ArXiv e-prints, 2015.
[36] Miller, J. and Sheffield, S., ‘Gaussian free field light cones and SLE $_{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D70C})$ ’. ArXiv e-prints, 2016.
[37] Miller, J. and Sheffield, S., ‘Imaginary geometry I: interacting SLEs’, Probab. Theory Related Fields 164(3–4) (2016), 553705.
[38] Miller, J. and Sheffield, S., ‘Imaginary geometry II: reversibility of SLE𝜅(𝜌1; 𝜌2) for 𝜅 ∈ (0, 4)’, Ann. Probab. 44(3) (2016), 16471722.
[39] Miller, J. and Sheffield, S., ‘Imaginary geometry III: reversibility of SLE𝜅 for 𝜅 ∈ (4, 8)’, Ann. of Math. (2) 184(2) (2016), 455486.
[40] Miller, J., Sheffield, S. and Werner, W., ‘Non-simple SLE curves are not determined by their range’. ArXiv e-prints, 2016.
[41] Miller, J., Sheffield, S. and Werner, W., ‘Conformal loop ensembles on Liouville quantum gravity’, 2017, in preparation.
[42] Miller, J., Sheffield, S. and Werner, W., ‘Labeled CLE interfaces and the Gaussian free field fan’, 2017, in preparation.
[43] Miller, J., Sun, N. and Wilson, D. B., ‘The Hausdorff dimension of the CLE gasket’, Ann. Probab. 42(4) (2014), 16441665.
[44] Miller, J. and Werner, W., ‘Connection probabilities for conformal loop ensembles’. ArXiv e-prints, 2017.
[45] Nacu, Ş. and Werner, W., ‘Random soups, carpets and fractal dimensions’, J. Lond. Math. Soc. (2) 83(3) (2011), 789809.
[46] Revuz, D. and Yor, M., Continuous Martingales and Brownian Motion, third edition, Grundlehren der Mathematischen Wissenschaften, 293 [Fundamental Principles of Mathematical Sciences] (Springer, Berlin, 1999).
[47] Rohde, S. and Schramm, O., ‘Basic properties of SLE’, Ann. of Math. (2) 161(2) (2005), 883924.
[48] Rozanov, Y. A., Markov Random Fields, Applications of Mathematics (Springer, New York–Berlin, 1982), Translated from the Russian by Constance M. Elson.
[49] Schramm, O., ‘Scaling limits of loop-erased random walks and uniform spanning trees’, Israel J. Math. 118 (2000), 221288.
[50] Schramm, O. and Sheffield, S., ‘Contour lines of the two-dimensional discrete Gaussian free field’, Acta Math. 202(1) (2009), 21137.
[51] Schramm, O. and Sheffield, S., ‘A contour line of the continuum Gaussian free field’, Probab. Theory Related Fields 157(1–2) (2013), 4780.
[52] Schramm, O., Sheffield, S. and Wilson, D. B., ‘Conformal radii for conformal loop ensembles’, Comm. Math. Phys. 288(1) (2009), 4353.
[53] Schramm, O. and Wilson, D. B., ‘SLE coordinate changes’, New York J. Math. 11 (2005), 659669. (electronic).
[54] Sepúlveda, A., ‘On thin local sets of the Gaussian free field’. ArXiv e-prints, 2017.
[55] Sheffield, S., ‘Exploration trees and conformal loop ensembles’, Duke Math. J. 147(1) (2009), 79129.
[56] Sheffield, S., ‘Conformal weldings of random surfaces: SLE and the quantum gravity zipper’, Ann. Probab. 44(5) (2016), 34743545.
[57] Sheffield, S., ‘Quantum gravity and inventory accumulation’, Ann. Probab. 44(6) (2016), 38043848.
[58] Sheffield, S. and Werner, W., ‘Conformal loop ensembles: the Markovian characterization and the loop-soup construction’, Ann. of Math. (2) 176(3) (2012), 18271917.
[59] Smirnov, S., ‘Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits’, C. R. Acad. Sci. Paris Sér. I Math. 333(3) (2001), 239244.
[60] Smirnov, S., ‘Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model’, Ann. of Math. (2) 172(2) (2010), 14351467.
[61] Smirnov, S., ‘Discrete complex analysis and probability’, inProceedings of the International Congress of Mathematicians, Vol. I (Hindustan Book Agency, New Delhi, 2010), 595621.
[62] Werner, W., Topics on the Two-dimensional Gaussian Free Field, Lecture Notes of ETH Graduate Course (2015).
[63] Werner, W. and Wu, H., ‘On conformally invariant CLE explorations’, Comm. Math. Phys. 320(3) (2013), 637661.
[64] Zhan, D., ‘Duality of chordal SLE’, Invent. Math. 174(2) (2008), 309353.
[65] Zhan, D., ‘Duality of chordal SLE, II’, Ann. Inst. Henri Poincaré Probab. Stat. 46(3) (2010), 740759.
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