On Multiple Schramm-Loewner Evolutions

TL;DR

This paper provides a probabilistic foundation for the ansatz of Multiple Schramm-Loewner Evolutions, showing it follows from conformal invariance and absolute continuity, and clarifies conditions on their parameters for consistency.

Contribution

It offers a probabilistic derivation of the multiple SLE ansatz and identifies specific parameter relations necessary for their consistent growth.

Findings

The ansatz is a consequence of conformal invariance and reparameterisation invariance.

Multiple SLEs are only consistent if their kappa parameters satisfy specific relations.

The relations are kappa_i = kappa_j or kappa_i = 16 / kappa_j.

Abstract

In this note we consider the ansatz for Multiple Schramm-Loewner Evolutions (SLEs) proposed by Bauer, Bernard and Kytola from a more probabilistic point of view. Here we show their ansatz is a consequence of conformal invariance, reparameterisation invariance and a notion of absolute continuity. In so doing we demonstrate that it is only consistent to grow multiple SLEs if their kappa parameters are related by kappa_i = kappa_j or kappa_i = 16 / kappa_j.