Numerical simulations of the spread from the mean of the SLE and multiple SLE dynamics

The Schramm-Loewner Evolution (SLE) describes a family of fractal curves that arise in the study of the scaling limits of many planar Statistical Physics models. These curves are modeled using the Loewner Differential Equation for the conformal maps [Formula: see text] with a Brownian motion driver. Using Euler’s Method, we performed numerical experiments to study the quantities [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the real part and [Formula: see text] refers to the sample average. These random variables measure the “spread” of the dynamics from the average behavior at fixed time. In the SLE case, our experiments predict that the distribution is bimodal when the dynamics started close to the origin and can become bell-shaped if the dynamics is started further from the origin. We also performed experiments for a Multiple SLE model whose driver is Dyson Brownian Motion. Due to singularity in the dynamics of the drivers and the many data points needed, this part is challenging from a computational perspective. In the Multiple SLE case, our experiments predict that the distribution is bell-shaped in all cases. We end by discussing the implications of our findings on the study of Multiple SLE traces/hulls along with future directions.