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Imaginary geometry. III: Reversibility of $$\mathrm{SLE}_\kappa$$ for $$\kappa \in (4,8)$$. (English) Zbl 1393.60092
From the text: Fix $$k \in (2,4)$$, and write $$k' = {{16} \mathord{\left/ {\vphantom {{16} k}} \right. \kern-\nulldelimiterspace} k} \in (4,8)$$. Our main result is the following:
Theorem 1.1. Suppose that $$D$$ is a Jordan domain, and let $$x,y \in \partial D$$ be distinct. Let $$\eta '$$ be a chordal $$\mathrm{SLE}_{k'}$$ process in $$D$$ from $$x$$ to $$y$$. Then the law of $$\eta '$$ has time-reversal symmetry. That is, if $$\psi :D \to D$$ is an anti-conformal map that swaps $$x$$ and $$y$$, then the time-reversal of $$\psi \circ \eta '$$ is equal in law to $$\eta '$$, up to reparametrization.
Theorem 1.1 is a special case of a more general theorem that gives the time-reversal symmetry of $$\mathrm{SLE}_{k'}({\rho _1},{\rho _2})$$ processes provided $${\rho _1},{\rho _2} \geqslant {{k'} \mathord{\left/ {\vphantom {{k'} 2}} \right. \kern-\nulldelimiterspace} 2} - 4$$.
Theorem 1.2. Suppose that $$D$$ is a Jordan domain, and let $$x,y \in \partial D$$ be distinct. Suppose that $$\eta '$$ is a chordal $$\mathrm{SLE}_{k'}({\rho _1},{\rho _2})$$ process in $$D$$ from $$x$$ to $$y$$ where the force points are located at $${x^ - }$$ and $${x^ + }$$. If $$\psi :D \to D$$ is an anti-conformal map that swaps $$x$$ and $$y$$, then the time-reversal of $$\psi \circ \eta '$$ is an $$\mathrm{SLE}_{k'}({\rho _1},{\rho _2})$$ process from $$x$$ to $$y$$, up to reparametrization.
Our final result is the nonreversibility of $$\mathrm{SLE}_{k'}({\rho _1},{\rho _2})$$ processes when either $${\rho _1}<{{k'} \mathord{\left/ {\vphantom {{k'} 2}} \right. \kern-\nulldelimiterspace} 2} - 4$$ or $${\rho _2} <{{k'} \mathord{\left/ {\vphantom {{k'} 2}} \right. \kern-\nulldelimiterspace} 2} - 4$$.
Theorem 3.1. Suppose that $$D$$ is a Jordan domain, and let $$x,y \in \partial D$$ be distinct. Suppose that $$\eta '$$ is a chordal $$\mathrm{SLE}_{k'}({\rho _1},{\rho _2})$$ process in $$D$$ from $$x$$ to $$y$$. Let $$\psi :D \to D$$ be an anti-conformal map that swaps $$x$$ and $$y$$. If either $${\rho _1} < {{k'} \mathord{\left/ {\vphantom {{k'} 2}} \right. \kern-\nulldelimiterspace} 2} - 4$$ or $${\rho _2} <{{k'} \mathord{\left/ {\vphantom {{k'} 2}} \right. \kern-\nulldelimiterspace} 2} - 4$$, then the law of the time-reversal of $$\psi (\eta ')$$ is not an $$\mathrm{SLE}_{k'}(\rho )$$ process for any collection of weights $$\rho$$.
For Part I and Part II see [the authors, Probab. Theory Relat. Fields 164, No. 3–4, 553–705 (2016; Zbl 1336.60162); Ann. Probab. 44, No. 3, 1647–1722 (2016; Zbl 1344.60078)].

##### MSC:
 60J67 Stochastic (Schramm-)Loewner evolution (SLE) 60G60 Random fields 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G15 Gaussian processes 60D05 Geometric probability and stochastic geometry
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##### References:
 [1] G. Ben Arous and J. -D. Deuschel, ”The construction of the $$(d+1)$$-dimensional Gaussian droplet,” Comm. Math. Phys., vol. 179, iss. 2, pp. 467-488, 1996. · Zbl 0858.60096 [2] F. Camia and C. M. Newman, ”Two-dimensional critical percolation: the full scaling limit,” Comm. Math. Phys., vol. 268, iss. 1, pp. 1-38, 2006. · Zbl 1117.60086 [3] D. Chelkak and S. Smirnov, ”Universality in the 2D Ising model and conformal invariance of fermionic observables,” Invent. Math., vol. 189, iss. 3, pp. 515-580, 2012. · Zbl 1257.82020 [4] J. Dubédat, ”Commutation relations for Schramm-Loewner evolutions,” Comm. Pure Appl. Math., vol. 60, iss. 12, pp. 1792-1847, 2007. · Zbl 1137.82009 [5] J. Dubédat, ”Duality of Schramm-Loewner evolutions,” Ann. Sci. Éc. Norm. Supér., vol. 42, iss. 5, pp. 697-724, 2009. · Zbl 1205.60147 [6] J. Dubédat, ”SLE and the free field: partition functions and couplings,” J. Amer. Math. Soc., vol. 22, iss. 4, pp. 995-1054, 2009. · Zbl 1204.60079 [7] C. Hagendorf, D. Bernard, and M. Bauer, ”The Gaussian free field and $${ SLE}_4$$ on doubly connected domains,” J. Stat. Phys., vol. 140, iss. 1, pp. 1-26, 2010. · Zbl 1193.82027 [8] K. Izyurov and K. Kytölä, ”Hadamard’s formula and couplings of SLEs with free field,” Probab. Theory Related Fields, vol. 155, iss. 1-2, pp. 35-69, 2013. · Zbl 1269.60067 [9] R. Kenyon, ”Dominos and the Gaussian free field,” Ann. Probab., vol. 29, iss. 3, pp. 1128-1137, 2001. · Zbl 1034.82021 [10] G. F. Lawler, Conformally Invariant Processes in the Plane, Providence, RI: Amer. Math. Soc., 2005, vol. 114. · Zbl 1074.60002 [11] G. F. Lawler, O. Schramm, and W. Werner, ”Conformal invariance of planar loop-erased random walks and uniform spanning trees,” Ann. Probab., vol. 32, iss. 1B, pp. 939-995, 2004. · Zbl 1126.82011 [12] J. Miller, Universality for SLE(4), 2010. [13] J. Miller, ”Fluctuations for the Ginzburg-Landau $$\nabla\phi$$ interface model on a bounded domain,” Comm. Math. Phys., vol. 308, iss. 3, pp. 591-639, 2011. · Zbl 1237.82030 [14] N. Makarov and S. Smirnov, ”Off-critical lattice models and massive SLEs,” in XVIth International Congress on Mathematical Physics, Hackensack, NJ: World Sci. Publ., 2010, pp. 362-371. · Zbl 1205.82055 [15] J. Miller and S. Sheffield, Imaginary geometry I: Interacting SLEs, 2012. · Zbl 1336.60162 [16] J. Miller and S. Sheffield, Imaginary geometry II: Reversibility of SLE$$_\kappa(\rho_1;\rho_2)$$ for $$\kappa \in (0,4)$$, 2012. · Zbl 1344.60078 [17] A. Naddaf and T. Spencer, ”On homogenization and scaling limit of some gradient perturbations of a massless free field,” Comm. Math. Phys., vol. 183, iss. 1, pp. 55-84, 1997. · Zbl 0871.35010 [18] S. Rohde and O. Schramm, ”Basic properties of SLE,” Ann. of Math., vol. 161, iss. 2, pp. 883-924, 2005. · Zbl 1081.60069 [19] B. Rider and B. Virág, ”The noise in the circular law and the Gaussian free field,” Int. Math. Res. Not., vol. 2007, iss. 2, p. I, 2007. · Zbl 1130.60030 [20] O. Schramm, ”Scaling limits of loop-erased random walks and uniform spanning trees,” Israel J. Math., vol. 118, pp. 221-288, 2000. · Zbl 0968.60093 [21] O. Schramm, ”A percolation formula,” Electron. Comm. Probab., vol. 6, pp. 115-120, 2001. · Zbl 1008.60100 [22] S. Sheffield, Local sets of the Gaussian free field: slides and audio. · Zbl 1132.60072 [23] S. Sheffield, ”Exploration trees and conformal loop ensembles,” Duke Math. J., vol. 147, iss. 1, pp. 79-129, 2009. · Zbl 1170.60008 [24] S. Sheffield, Conformal weldings of random surfaces: SLE and the quantum gravity zipper, 2010. · Zbl 1388.60144 [25] S. Smirnov, ”Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits,” C. R. Acad. Sci. Paris Sér. I Math., vol. 333, iss. 3, pp. 239-244, 2001. · Zbl 0985.60090 [26] S. Smirnov, ”Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model,” Ann. of Math., vol. 172, iss. 2, pp. 1435-1467, 2010. · Zbl 1200.82011 [27] O. Schramm and S. Sheffield, ”Harmonic explorer and its convergence to $${ SLE}_4$$,” Ann. Probab., vol. 33, iss. 6, pp. 2127-2148, 2005. · Zbl 1095.60007 [28] O. Schramm and S. Sheffield, ”Contour lines of the two-dimensional discrete Gaussian free field,” Acta Math., vol. 202, iss. 1, pp. 21-137, 2009. · Zbl 1210.60051 [29] O. Schramm and S. Sheffield, ”A contour line of the continuum Gaussian free field,” Probab. Theory Related Fields, vol. 157, iss. 1-2, pp. 47-80, 2013. · Zbl 1331.60090 [30] S. Sheffield and W. Werner, ”Conformal loop ensembles: the Markovian characterization and the loop-soup construction,” Ann. of Math., vol. 176, iss. 3, pp. 1827-1917, 2012. · Zbl 1271.60090 [31] W. Werner, ”Random planar curves and Schramm-Loewner evolutions,” in Lectures on Probability Theory and Statistics, New York: Springer-Verlag, 2004, vol. 1840, pp. 107-195. · Zbl 1057.60078 [32] D. Zhan, ”Reversibility of chordal SLE,” Ann. Probab., vol. 36, iss. 4, pp. 1472-1494, 2008. · Zbl 1157.60051 [33] D. Zhan, ”Reversibility of some chordal $${ SLE}(\kappa;\rho)$$ traces,” J. Stat. Phys., vol. 139, iss. 6, pp. 1013-1032, 2010. · Zbl 1205.82063
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