Numerical studies of planar closed random walks

TL;DR

This paper presents lattice numerical simulations of planar closed random walks, analyzing their frontiers and winding sectors, revealing a Hausdorff dimension of approximately 1.77 when considering inner sectors, which refines previous estimates.

Contribution

It provides a detailed numerical analysis of the Hausdorff dimensions of the frontiers and winding sectors of planar closed random walks, including the effect of inner sectors.

Findings

Frontiers have Hausdorff dimension 4/3.

Inner 0-winding sectors increase frontier dimension to approximately 1.77.

Numerical simulations support theoretical predictions.

Abstract

Lattice numerical simulations for planar closed random walks and their winding sectors are presented. The frontiers of the random walks and of their winding sectors have a Hausdorff dimension $d_H=4/3$. However, when properly defined by taking into account the inner 0-winding sectors, the frontiers of the random walks have a Hausdorff dimension $d_H\approx 1.77$.