Iterative Schwarz-Christoffel Transformations Driven by Random Walks and Fractal Curves

TL;DR

This paper introduces an iterative Schwarz-Christoffel transformation method driven by random walks to generate fractal curves, demonstrating convergence to SLE curves and providing a new approach to simulate complex conformal mappings.

Contribution

It presents a novel iterative SCT approach driven by binomial stochastic processes that approximates SLE curves, linking conformal maps with stochastic path generation.

Findings

Rescaled ISCT paths exhibit statistical properties similar to SLE curves.

Numerical evidence supports convergence of ISCT paths to SLE in the limit.

The method provides a new way to simulate fractal conformal boundaries.

Abstract

Stochastic Loewner evolution (SLE) is a differential equation driven by a one-dimensional Brownian motion (BM), whose solution gives a stochastic process of conformal transformation on the upper half complex-plane $\H$. As an evolutionary boundary of image of the transformation, a random curve (the SLE curve) is generated, which is starting from the origin and running in $\H$ toward the infinity as time is going. The SLE curves provides a variety of statistical ensembles of important fractal curves, if we change the diffusion constant of the driving BM. In the present paper, we consider the Schwarz-Christoffel transformation (SCT), which is a conformal map from $\H$ to the region $\H$ with a slit starting from the origin. We prepare a binomial system of SCTs, one of which generates a slit in $\H$ with an angle $\alpha \pi$ from the positive direction of the real axis, and the other of which with an angle $(1-\alpha) \pi$. One parameter $\kappa >0$ is introduced to control the value of $\alpha$ and the length of slit. Driven by a one-dimensional random walk, which is a binomial stochastic process, a random iteration of SCTs is performed. By interpolating tips of slits by straight lines, we have a random path in $\H$, which we call an Iterative SCT (ISCT) path. It is well-known that, as the number of steps $N$ of random walk goes infinity, each path of random walk divided by $\sqrt{N}$ converges to a Brownian curve. Then we expect that the ISCT paths divided by $\sqrt{N}$ (the rescaled ISCT paths) converge to the SLE curves in $N \to \infty$. Our numerical study implies that, for sufficiently large $N$, the rescaled ISCT paths will have the same statistical properties as the SLE curves have, supporting our expectation.