Numerical computations for the Schramm-Loewner Evolution

TL;DR

This paper reviews numerical methods for simulating and analyzing Schramm-Loewner Evolution (SLE) curves, including algorithms for generating curves and testing if given curves are SLE, with a focus on practical implementation and open problems.

Contribution

It provides a pedagogic review of two numerical methods related to SLE, including simulation and inverse algorithms, highlighting practical issues and open challenges.

Findings

Algorithms for simulating SLE curves demonstrated.

Methods for testing if curves are SLE via driving function analysis.

Discussion of practical issues and open problems in numerical SLE computations.

Abstract

We review two numerical methods related to the Schramm-Loewner evolution (SLE). The first simulates SLE itself. More generally, it finds the curve in the half-plane that results from the Loewner equation for a given driving function. The second method can be thought of as the inverse problem. Given a simple curve in the half-plane it computes the driving function in the Loewner equation. This algorithm can be used to test if a given random family of curves in the half-plane is SLE by computing the driving process for the curves and testing if it is Brownian motion. More generally, this algorithm can be used to compute the driving process for random curves that may not be SLE. Most of the material presented here has appeared before. Our goal is to give a pedagogic review, illustrate some of the practical issues that arise in these computations and discuss some open problems.