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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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