Open and closed random walks with fixed edgelengths in $\mathbb{R}^d$

TL;DR

This paper introduces a geometric median-based method to approximate the closest closed equilateral random walk to any open one in ^d, analyzes its probabilistic properties, and explores implications for knotting in polygons.

Contribution

It provides an explicit construction for closing open equilateral polygons and establishes probabilistic convergence results for the closure distance in high dimensions.

Findings

Closure distance converges to a Nakagami distribution as n 

Probabilistic bounds on the closure distance for fixed edgelengths

Numerical evidence suggests the closure map preserves natural measures

Abstract

In this paper, we consider fixed edgelength $n$-step random walks in $\mathbb{R}^d$. We give an explicit construction for the closest closed equilateral random walk to almost any open equilateral random walk based on the geometric median, providing a natural map from open polygons to closed polygons of the same edgelength. Using this, we first prove that a natural reconfiguration distance to closure converges in distribution to a Nakagami$(\frac{d}{2},\frac{d}{d-1})$ random variable as $n \rightarrow \infty$. We then strengthen this to an explicit probabilistic bound on the distance to closure for a random $n$-gon in any dimension with any collection of fixed edgelengths $w_i$. Numerical evidence supports the conjecture that our closure map pushes forward the natural probability measure on open polygons to something very close to the natural probability measure on closed polygons; if this is so, we can draw some conclusions about the frequency of local knots in closed polygons of fixed edgelength.