SLE Animation

Explanation

The green particle peforms a Brownian motion of diffusivity κ = 2 on the real line, while the blue particles in the upper half plane move in the Loewner flow. This means that each particle is subjected to a force given by the vector field shown below, changing with time so that it's always centered at the location of the green particle. When the green particle collides with a blue particle, the original location of the blue particle is appended to the SLE curve, which is shown in red on a second copy of the upper half-plane.

[Loewner Field]

Disclaimer

The purpose of the animation is only to illustrate how the Loewner transform works. In particular, the path obtained with this approach bears little resemblance to an SLE path. This deficiency stems from the discretization's implicit assumption that a near-collision of the green particle with a blue particle is about the same as an actual collision with a nearby blue particle. This assumption yields poor results because of the regularity of the conformal maps breaks down near the fractal SLE curve. For a numerical approach better adapted to the goal of producing representative paths, see Tom Kennedy's paper on numerical SLE methods.