Open and closed random walks with fixed edgelengths in ℝd

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 (d2dd-1)random variable as n → ∞. 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.