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CLE percolations. (English) Zbl 1390.60356
Summary: 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 (CLE\(_\kappa\) for \(\kappa\in (8/3,4)\), whose loops are Schramm’s SLE\(_\kappa\)-type curves) and the corresponding CLEs with nonsimple loops (CLE\(_{\kappa^{\prime}}\) with \(\kappa^{\prime}:=16/\kappa\in (4,6)\), whose loops are SLE\(_{\kappa^{\prime}}\)-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 \(\kappa\in (8/3,4)\), one can construct a variant of CLE\(_\kappa\) as follows: start with an instance of CLE\(_{\kappa^{\prime}}\), then use a biased coin to independently color each CLE\(_{\kappa^{\prime}}\) 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 CLE\(_{\kappa^{\prime}}\) loops as interfaces of a continuum analog of critical Bernoulli percolation within CLE\(_\kappa\) 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 SLE\(_6\) and 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 SLE\(_\kappa(\rho)\) curves for \(\rho<-2\), such as their decomposition into collections of SLE\(_\kappa\)-type ‘loops’ hanging off of SLE\(_{\kappa^{\prime}}\)-type ‘trunks’, and vice versa (exchanging \(\kappa\) and \(\kappa^{\prime}\)). We also define a continuous family of natural CLE variants called boundary conformal loop ensembles (BCLEs) that share some (but not all) of the conformal symmetries that characterize CLEs, and that should be scaling limits of critical models with special boundary conditions. We extend the CLE\(_\kappa\)/CLE\(_{\kappa^{\prime}}\) correspondence to a BCLE\(_\kappa\)/BCLE\(_{\kappa^{\prime}}\) correspondence that makes sense for the wider range \(\kappa\in(2,4]\) and \(\kappa^{\prime}\in [4,8)\).

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
82B43 Percolation
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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, 1421-1452, (2008) · Zbl 1165.60007
[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), 528-571. doi:10.1214/13-AOP881 · Zbl 1332.82012
[4] Camia, F. and Newman, C. M., ‘Two-dimensional critical percolation: the full scaling limit’, Comm. Math. Phys.268(1) (2006), 1-38. doi:10.1007/s00220-006-0086-1 · Zbl 1117.60086
[5] Cardy, J., ‘Conformal field theory and statistical mechanics’, inExact Methods in Low-dimensional Statistical Physics and Quantum Computing (Oxford University Press, Oxford, 2010), 65-98. · Zbl 1202.82011
[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. Paris352(2) (2014), 157-161. doi:10.1016/j.crma.2013.12.002 · Zbl 1294.82007
[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), 1087-1138. doi:10.4007/annals.2015.181.3.5 · Zbl 1318.82006
[8] Chelkak, D. and Smirnov, S., ‘Universality in the 2D Ising model and conformal invariance of fermionic observables’, Invent. Math.189(3) (2012), 515-580. doi:10.1007/s00222-011-0371-2 · Zbl 1257.82020
[9] Dubédat, J., Commutation relations for Schramm-Loewner evolutions, Comm. Pure Appl. Math., 60, 12, 1792-1847, (2007) · Zbl 1137.82009
[10] Dubédat, J., Duality of Schramm-Loewner evolutions, Ann. Sci. Éc. Norm. Supér. (4), 42, 5, 697-724, (2009) · Zbl 1205.60147
[11] Dubédat, J., SLE and the free field: partition functions and couplings, J. Amer. Math. Soc., 22, 4, 995-1054, (2009) · Zbl 1204.60079
[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), 2009-2012. doi:10.1103/PhysRevD.38.2009
[15] Fortuin, C. M. and Kasteleyn, P. W., ‘On the random-cluster model. I. Introduction and relation to other models’, Physica57 (1972), 536-564. doi:10.1016/0031-8914(72)90045-6
[16] Grimmett, G., The Random-Cluster Model, (2006), Springer: Springer, Berlin · Zbl 1122.60087
[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. · Zbl 1415.60033
[18] Gwynne, E. and Miller, J., ‘Convergence of the topology of critical Fortuin-Kasteleyn planar maps to that of CLE\(_{𝜅}\) 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, 1149-1159, (1999) · Zbl 0957.60099
[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), 1-24. · Zbl 1169.82006
[23] Hongler, C. and Kytölä, K., ‘Ising interfaces and free boundary conditions’, J. Amer. Math. Soc.26(4) (2013), 1107-1189. doi:10.1090/S0894-0347-2013-00774-2 · Zbl 1284.82021
[24] Hongler, C. and Smirnov, S., ‘The energy density in the planar Ising model’, Acta Math.211(2) (2013), 191-225. doi:10.1007/s11511-013-0102-1 · Zbl 1287.82007
[25] Izyurov, K., Smirnov’s observable for free boundary conditions, interfaces and crossing probabilities, Comm. Math. Phys., 337, 1, 225-252, (2015) · Zbl 1318.82010
[26] Kemppainen, A. and Smirnov, S., ‘Conformal invariance of boundary touching loops of FK Ising model’. ArXiv e-prints, 2015. · Zbl 1422.60140
[27] Kemppainen, A. and S., Smirnov, ‘Random curves, scaling limits and Loewner evolutions’, Ann. Probab.45(2) (2017), 698-779. doi:10.1214/15-AOP1074 · Zbl 1393.60016
[28] Kumagai, T., ‘Percolation on pre-Sierpinski carpets’, inNew Trends in Stochastic Analysis (Charingworth, 1994) (World Sci. Publ., River Edge, NJ, 1997), 288-304.
[29] Lawler, G., Conformally Invariant Processes in the Plane, (2005), American Mathematical Society: American Mathematical Society, Providence, RI · Zbl 1074.60002
[30] Lawler, G., Schramm, O. and Werner, W., ‘Conformal restriction: the chordal case’, J. Amer. Math. Soc.16(4) (2003), 917-955. (electronic). doi:10.1090/S0894-0347-03-00430-2 · Zbl 1030.60096
[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), 939-995. doi:10.1214/aop/1079021469 · Zbl 1126.82011
[32] Maes, C. and Vande Velde, K., ‘The fuzzy Potts model’, J. Phys. A28(15) (1995), 4261-4270. doi:10.1088/0305-4470/28/15/007 · Zbl 0868.60081
[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. · Zbl 1378.60108
[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\(_{\unicode[STIX]{x1D705}}(𝜌)\)’. ArXiv e-prints, 2016.
[37] Miller, J. and Sheffield, S., ‘Imaginary geometry I: interacting SLEs’, Probab. Theory Related Fields164(3-4) (2016), 553-705. doi:10.1007/s00440-016-0698-0 · Zbl 1336.60162
[38] Miller, J. and Sheffield, S., ‘Imaginary geometry II: reversibility of SLE_{𝜅}(𝜌_{1}; 𝜌_{2}) for 𝜅 ∈ (0, 4)’, Ann. Probab.44(3) (2016), 1647-1722. doi:10.1214/14-AOP943 · Zbl 1344.60078
[39] Miller, J. and Sheffield, S., ‘Imaginary geometry III: reversibility of SLE_{𝜅} for 𝜅 ∈ (4, 8)’, Ann. of Math. (2)184(2) (2016), 455-486. doi:10.4007/annals.2016.184.2.3 · Zbl 1393.60092
[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), 1644-1665. doi:10.1214/12-AOP820 · Zbl 1305.60078
[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), 789-809. doi:10.1112/jlms/jdq094 · Zbl 1223.28012
[46] Revuz, D. and Yor, M., Continuous Martingales and Brownian Motion, third edition, (Springer, Berlin, 1999). doi:10.1007/978-3-662-06400-9 · Zbl 0917.60006
[47] Rohde, S. and Schramm, O., ‘Basic properties of SLE’, Ann. of Math. (2)161(2) (2005), 883-924. doi:10.4007/annals.2005.161.883 · Zbl 1081.60069
[48] Rozanov, Y. A., Markov Random Fields, (Springer, New York-Berlin, 1982), Translated from the Russian by Constance M. Elson. doi:10.1007/978-1-4613-8190-7
[49] Schramm, O., Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math., 118, 221-288, (2000) · Zbl 0968.60093
[50] Schramm, O. and Sheffield, S., ‘Contour lines of the two-dimensional discrete Gaussian free field’, Acta Math.202(1) (2009), 21-137. doi:10.1007/s11511-009-0034-y · Zbl 1210.60051
[51] Schramm, O. and Sheffield, S., ‘A contour line of the continuum Gaussian free field’, Probab. Theory Related Fields157(1-2) (2013), 47-80. doi:10.1007/s00440-012-0449-9 · Zbl 1331.60090
[52] Schramm, O., Sheffield, S. and Wilson, D. B., ‘Conformal radii for conformal loop ensembles’, Comm. Math. Phys.288(1) (2009), 43-53. doi:10.1007/s00220-009-0731-6 · Zbl 1187.82044
[53] Schramm, O. and Wilson, D. B., ‘SLE coordinate changes’, New York J. Math.11 (2005), 659-669. (electronic). · Zbl 1094.82007
[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, 79-129, (2009) · Zbl 1170.60008
[56] Sheffield, S., Conformal weldings of random surfaces: SLE and the quantum gravity zipper, Ann. Probab., 44, 5, 3474-3545, (2016) · Zbl 1388.60144
[57] Sheffield, S., Quantum gravity and inventory accumulation, Ann. Probab., 44, 6, 3804-3848, (2016) · Zbl 1359.60120
[58] Sheffield, S. and Werner, W., ‘Conformal loop ensembles: the Markovian characterization and the loop-soup construction’, Ann. of Math. (2)176(3) (2012), 1827-1917. doi:10.4007/annals.2012.176.3.8 · Zbl 1271.60090
[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, 239-244, (2001) · Zbl 0985.60090
[60] Smirnov, S., Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model, Ann. of Math. (2), 172, 2, 1435-1467, (2010) · Zbl 1200.82011
[61] Smirnov, S., ‘Discrete complex analysis and probability’, inProceedings of the International Congress of Mathematicians, Vol. I (Hindustan Book Agency, New Delhi, 2010), 595-621. · Zbl 1251.30049
[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), 637-661. doi:10.1007/s00220-013-1719-9 · Zbl 1290.60082
[64] Zhan, D., Duality of chordal SLE, Invent. Math., 174, 2, 309-353, (2008) · Zbl 1158.60047
[65] Zhan, D., Duality of chordal SLE, II, Ann. Inst. Henri Poincaré Probab. Stat., 46, 3, 740-759, (2010) · Zbl 1200.60071
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