On radial stochastic Loewner evolution in multiply connected domains

TL;DR

This paper extends radial SLE to multiply connected domains, establishing a new Loewner equation, analyzing the associated moduli space dynamics, and characterizing the diffusion processes that generate conformally invariant random curves.

Contribution

It introduces the radial Komatu-Loewner equation for multiply connected domains and characterizes the diffusion processes describing conformally invariant curves.

Findings

The vector field for moduli motion is Lipschitz.

Conformally invariant curves are described by multidimensional diffusions.

Locality property holds only for =6.

Abstract

We discuss the extension of radial SLE to multiply connected planar domains. First, we extend Loewner's theory of slit mappings to multiply connected domains by establishing the radial Komatu-Loewner equation, and show that a simple curve from the boundary to the bulk is encoded by a motion on moduli space and a motion on the boundary of the domain. Then, we show that the vector-field describing the motion of the moduli is Lipschitz. We explain why this implies that "consistent," conformally invariant random simple curves are described by multidimensional diffusions, where one component is a motion on the boundary, and the other component is a motion on moduli space. We argue what the exact form of this diffusion is (up to a single real parameter $\kappa$) in order to model boundaries of percolation clusters. Finally, we show that this moduli diffusion leads to random non-self-crossing curves satisfying the locality property if and only if $\kappa=6$.