Defining SLE in multiply connected domains with the Brownian loop measure

TL;DR

This paper extends the definition of Schramm-Loewner evolution (SLE) to multiply connected domains using the Brownian loop measure, providing new insights and derivations for the annulus case.

Contribution

It introduces a novel approach to defining SLE in multiply connected domains via the Brownian loop measure, connecting to recent work by Zhan.

Findings

SLE in multiply connected domains is defined for  f  using the Brownian loop measure.

The measure in the annulus matches recent results by Zhan.

A new derivation of the PDE for the partition function in the annulus is provided.

Abstract

We define the Schramm-Loewner evolution (SLE) in multiply connected domains for kappa \leq 4 using the Brownian loop measure. We show that in the case of the annulus, this is the same measure obtained recently by Dapeng Zhan. We use the loop formulation to give a different derivation of the partial differential equation for the partition function for the annulus.