Duality of Chordal SLE

Abstract

We derive some geometric properties of chordal 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 SLE$(\kappa ;\vec{\rho })$ process has the same distribution as the image of a chordal SLE$({\kappa }^{\prime };\overrightarrow{{\rho }^{\prime }})$ trace, where $\kappa >4$, ${\kappa }^{\prime }=16/\kappa$, and the forces $\vec{\rho }$ and $\overrightarrow{{\rho }^{\prime }}$ are suitably chosen. We find that for $\kappa \ge 8$, the boundary of a standard chordal SLE$(\kappa )$ hull stopped on swallowing a fixed $x\in\R\sem\{0\}$ is the image of some SLE$(16/\kappa ;\vec{\rho })$ trace started from $x$. Then we obtain a new proof of the fact that chordal SLE$(\kappa )$ trace is not reversible for $\kappa >8$. We also prove that the reversal of SLE$(4;\vec{\rho })$ trace has the same distribution as the time-change of some SLE$(4;\overrightarrow{{\rho }^{\prime }})$ trace for certain values of $\vec{\rho }$ and $\overrightarrow{{\rho }^{\prime }}$.

In this third version, the referee's suggestions are taken into consideration. More details are added. Some typos are corrected. The paper has been accepted by Inventiones Mathematicae

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• The dimension of the SLE curves
• Conformal restriction: the chordal case
• SLE coordinate changes
• Commutation relations for SLE
• Reversibility of chordal SLE
• Dipolar SLEs
• SLE(κ,ρ) martingales and duality
• The scaling limits of planar LERW in finitely connected domains
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