Irregular traces of multiple SLE(0) systems with multiple marked points

TL;DR

This paper investigates the geometric behavior of multiple SLE(0) traces with multiple marked points, revealing regularity properties, effects of spin, and complex behaviors like spiraling, supported by explicit MATLAB examples.

Contribution

It characterizes the regularity and spiraling behavior of multiple SLE(0) traces with multiple marked points, including effects of spin and counterexamples.

Findings

Trajectories are regular near singularities in chordal and radial cases.

Spin induces spiraling at interior marked points in the radial case.

Multiple interior points can cause asymptotic convergence and spiraling, breaking regularity.

Abstract

In this supplementary note, we study the traces of multiple SLE(0) systems with two or more additional marked points. For general chordal configurations, the traces correspond to the real locus of real rational functions; in the radial case, they correspond to the horizontal trajectories of residue-free quadratic differentials. In both settings, we establish the regularity of the trajectories near singularities: no spiraling occurs, and no two trajectories asymptotically converge to the same direction. Moreover, in the radial case with non-zero spin at the marked interior point, we show that the spin induces a spiraling behavior at the marked interior point. However, this regularity breaks down when multiple interior marked points are present. In such cases, trajectories may asymptotically approach the same direction, and spiraling can occur even in the absence of spin. We present explicit counterexamples generated using MATLAB, with code provided for reference.