SLE on doubly-connected domains and the winding of loop-erased random walks

TL;DR

This paper investigates properties of SLE traces on doubly-connected domains, specifically cylinders, revealing endpoint and winding probabilities for SLE_2 and endpoint distributions for SLE_4, with connections to Brownian motion.

Contribution

It provides new analytical results for SLE on doubly-connected domains, including endpoint and winding distributions, and links these to classical stochastic processes.

Findings

Derived endpoint and winding probabilities for SLE_2 on a cylinder.

Obtained endpoint distribution for SLE_4 on a cylinder.

Established connections between SLE properties and Brownian motion.

Abstract

Two-dimensional loop-erased random walks (LERWs) are random planar curves whose scaling limit is known to be a Schramm-Loewner evolution SLE_k with parameter k = 2. In this note, some properties of an SLE_k trace on doubly-connected domains are studied and a connection to passive scalar diffusion in a Burgers flow is emphasised. In particular, the endpoint probability distribution and winding probabilities for SLE_2 on a cylinder, starting from one boundary component and stopped when hitting the other, are found. A relation of the result to conditioned one-dimensional Brownian motion is pointed out. Moreover, this result permits to study the statistics of the winding number for SLE_2 with fixed endpoints. A solution for the endpoint distribution of SLE_4 on the cylinder is obtained and a relation to reflected Brownian motion pointed out.