On Chordal and Bilateral SLE in multiply connected domains

TL;DR

This paper extends Loewner's equation to multiply connected domains to analyze conformally invariant random curves, identifying conditions under which these curves exhibit Markovian properties and locality.

Contribution

It introduces a framework for studying conformally invariant curves in multiply connected domains, extending Loewner's equation and characterizing possible random curve candidates.

Findings

Markovian boundary motion when combined with moduli dynamics

Candidates labeled by a real constant and a homogeneous function

Locality property holds when interaction vanishes and parameter equals six

Abstract

We discuss the possible candidates for conformally invariant random non-self-crossing curves which begin and end on the boundary of a multiply connected planar domain, and which satisfy a Markovian-type property. We consider both, the case when the curve connects a boundary component to itself (chordal), and the case when the curve connects two different boundary components (bilateral). We establish appropriate extensions of Loewner's equation to multiply connected domains for the two cases. We show that a curve in the domain induces a motion on the boundary and that this motion is enough to first recover the motion of the moduli of the domain and then, second, the curve in the interior. For random curves in the interior we show that the induced random motion on the boundary is not Markov if the domain is multiply connected, but that the random motion on the boundary together with the random motion of the moduli forms a Markov process. In the chordal case, we show that this Markov process satisfies Brownian scaling and discuss how this limits the possible conformally invariant random non-self-crossing curves. We show that the possible candidates are labeled by a real constant and a function homogeneous of degree minus one which describes the interaction of the random curve with the boundary. We show that the random curve has the locality property if the interaction term vanishes and the real parameter equals six.