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Duality of chordal SLE. (English) Zbl 1158.60047
Author’s summary: We derive some geometric properties of chordal $$\text{SLE}(\kappa ;\vec{\rho})$$ processes. Using these results and the method of coupling two SLE processes, we prove that the outer boundary of the final hull of a chordal $$\text{SLE}(\kappa ;\vec{\rho})$$ process has the same distribution as the image of a chordal $$\text{SLE}(\kappa';\vec{\rho}\,'$$) trace, where $$\kappa >4, \kappa'=16/\kappa$$, and the forces $$\vec{\rho}$$ and $$\vec{\rho}\,'$$ are suitably chosen. We find that for $$\kappa \geq 8$$, the boundary of a standard chordal $$\text{SLE}(\kappa )$$ hull stopped on swallowing a fixed $$x\in\mathbb{R}\setminus\{0\}$$ is the image of some $$\text{SLE}(16/\kappa ;\vec{\rho})$$ trace started from $$x$$. Then we obtain a new proof of the fact that chordal $$\text{SLE}(\kappa )$$ trace is not reversible for $$\kappa >8$$. We also prove that the reversal of $$\text{SLE}(4;\vec{\rho})$$ trace has the same distribution as the time-change of some $$\text{SLE}(4;\vec{\rho}\,')$$ trace for certain values of $$\vec{\rho}$$ and $$\vec{\rho}\,'$$.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G99 Stochastic processes 60J65 Brownian motion 30C35 General theory of conformal mappings 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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