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Imaginary geometry. II: Reversibility of $$\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$$ for $$\kappa\in(0,4)$$. (English) Zbl 1344.60078
Summary: Given a simply connected planar domain $$D$$, distinct points $$x,y\in\partial D$$, and $$\kappa>0$$, the Schramm-Loewner evolution $$\operatorname{SLE}_{\kappa}$$ is a random continuous non-self-crossing path in $$\overline{D}$$ from $$x$$ to $$y$$. The $$\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$$ processes, defined for $$\rho_{1},\rho_{2}>-2$$, are in some sense the most natural generalizations of $$\operatorname{SLE}_{\kappa}$$.
When $$\kappa\leq 4$$, we prove that the law of the time-reversal of an $$\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$$ process from $$x$$ to $$y$$ is, up to parameterization, an $$\operatorname{SLE}_{\kappa}(\rho_{2};\rho_{1})$$ process from $$y$$ to $$x$$. This assumes that the “force points” used to define the $$\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$$ process are immediately to the left and right of the SLE seed. A generalization to arbitrary (and arbitrarily many) force points applies whenever the path does not (or is conditioned not to) hit $$\partial D\setminus\{x,y\}$$.
The proof of time-reversal symmetry makes use of the interpretation of $$\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$$ processes as a ray of a random geometry associated to the Gaussian free field. Within this framework, the time-reversal result allows us to couple two instances of the Gaussian free field (with different boundary conditions) so that their difference is almost surely constant on either side of the path. In a fairly general sense, adding appropriate constants to the two sides of a ray reverses its orientation.
Editorial remark: For Part I see [Probab. Theory Relat. Fields 164, No. 3–4, 553–705 (2016; Zbl 1336.60162)].

##### MSC:
 60J67 Stochastic (Schramm-)Loewner evolution (SLE) 60G15 Gaussian processes 60G60 Random fields 60D05 Geometric probability and stochastic geometry
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