There's a new paper (jointly with Jason Cantarella, Kyle Chapman, and Clayton Shonkwiler).
Open and closed random walks with fixed edgelengths in ℝd
In this paper, we consider fixed edgelength n-step random walks in ℝd. 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(d/2,d/(d−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 wi. 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.