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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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