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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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