Winding angle variance of Fortuin-Kasteleyn contours

TL;DR

This paper investigates the variance in winding numbers of various random fractal curves, focusing on typical points and complex strand interactions, providing evidence supporting a new conjecture.

Contribution

It introduces a novel approach to measure winding variance at typical points and complex strand junctions, extending previous endpoint-focused studies.

Findings

Measured winding variance aligns with the proposed conjecture.

Analysis includes multiple types of fractal curves and strand configurations.

Results suggest universal behavior across different models.

Abstract

The variance in the winding number of various random fractal curves, including the self-avoiding walk, the loop-erased random walk, contours of FK clusters, and stochastic Loewner evolution, have been studied by numerous researchers. Usually the focus has been on the winding at the endpoints. We measure the variance in winding number at typical points along the curve. More generally, we study the winding at points where k strands come together, and some adjacent strands may be conditioned not to hit each other. The measured values are consistent with an interesting conjecture.