Duality of Chordal SLE

TL;DR

This paper explores the geometric duality of chordal SLE processes, establishing distributional equivalences between different SLE traces and providing new insights into their reversibility and boundary properties.

Contribution

It introduces a duality relation between SLE$(\kappa; ho)$ processes and their images, and offers new proofs regarding the non-reversibility of SLE$(\kappa)$ for $\kappa>8$.

Findings

Outer boundary of SLE$(\kappa; ho)$ hulls has the same distribution as a dual SLE trace.

For $\kappa extgreater 8$, SLE hull boundary is an image of a dual SLE trace.

Chordal SLE$(\kappa)$ trace is not reversible for $\kappa>8$.

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$(\kappa';\vec{\rho'})$ trace, where $\kappa>4$, $\kappa'=16/\kappa$, and the forces $\vec{\rho}$ and $\vec{\rho'}$ are suitably chosen. We find that for $\kappa\ge 8$, 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$(4;\vec{\rho'})$ trace for certain values of $\vec{\rho}$ and $\vec{\rho'}$.