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

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.