Restriction Properties of Annulus SLE

TL;DR

This paper proves that annulus SLE$(;\Lambda)$ processes satisfy a restriction property similar to chordal SLE, enabling the construction of multiple crossing curves with specific SLE properties.

Contribution

It establishes a restriction property for annulus SLE$(;\Lambda)$ processes, expanding understanding of their geometric behavior and enabling new curve construction methods.

Findings

Annulus SLE$(;\Lambda)$ processes satisfy a restriction property.

Construction of $n\ge 2$ crossing curves in an annulus with SLE properties.

Last curve in the crossing configuration is a chordal SLE$()$ trace.

Abstract

For $\kappa\in(0,4]$, a family of annulus SLE$(\kappa;\Lambda)$ processes were introduced in [14] to prove the reversibility of whole-plane SLE$(\kappa)$. In this paper we prove that those annulus SLE$(\kappa;\Lambda)$ processes satisfy a restriction property, which is similar to that for chordal SLE$(\kappa)$. Using this property, we construct $n\ge 2$ curves crossing an annulus such that, when any $n-1$ curves are given, the last curve is a chordal SLE$(\kappa)$ trace.