Convergence of an algorithm simulating Loewner curves

TL;DR

This paper proves the convergence of a common algorithm for simulating Schramm--Loewner evolution (SLE) curves, ensuring accurate approximation of these fractal curves for various parameters, except at a critical value.

Contribution

It establishes a convergence condition for the algorithm simulating SLE curves, extending its validity across different regimes of the parameter , excluding =8.

Findings

The algorithm converges to SLE for all  except =8.

Provided a deterministic condition on driving functions for sup-norm convergence.

The convergence covers the regimes where SLE curves are simple, self-touching, and space-filling.

Abstract

The development of Schramm--Loewner evolution (SLE) as the scaling limits of discrete models from statistical physics makes direct simulation of SLE an important task. The most common method, suggested by Marshall and Rohde \cite{MR05}, is to sample Brownian motion at discrete times, interpolate appropriately in between and solve explicitly the Loewner equation with this approximation. This algorithm always produces piecewise smooth non self-intersecting curves whereas SLE$_\kappa$ has been proven to be simple for $\kappa\in[0,4]$, self-touching for $\kappa\in(4,8)$ and space-filling for $\kappa\geq 8$. In this paper we show that this sequence of curves converges to SLE$_\kappa$ for all $\kappa\neq 8$ by giving a condition on deterministic driving functions to ensure the sup-norm convergence of simulated curves when we use this algorithm.