Open and closed random walks with fixed edgelengths in Rd. (24th September 2018)

• Record Type: Journal Article
• Authors: Cantarella, Jason; Chapman, Kyle; Reiter, Philipp; Shonkwiler, Clayton
• Is Part Of: Journal of physics. Volume 51:Number 43(2018)
• Publication Date: 2018-09-24
• Subjects: Fermat–Weber problem -- geometric median -- random polygon -- random knot -- concentration of measure -- Nakagami distribution -- Bernstein inequality; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/aade0a
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: In this paper, we consider fixed edgelength n -step random walks in . 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 random variable as . 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.