The symplectic geometry of closed equilateral random walks in 3-space

Abstract

A closed equilateral random walk in 3-space is a selection of unit length vectors giving the steps of the walk conditioned on the assumption that the sum of the vectors is zero. The sample space of such walks with $n$ edges is the $(2n-3)$-dimensional Riemannian manifold of equilateral closed polygons in $\mathbb{R}^3$. We study closed random walks using the symplectic geometry of the $(2n-6)$-dimensional quotient of the manifold of polygons by the action of the rotation group $\operatorname {SO}(3)$. The basic objects of study are the moment maps on equilateral random polygon space given by the lengths of any $(n-3)$-tuple of nonintersecting diagonals. The Atiyah-Guillemin-Sternberg theorem shows that the image of such a moment map is a convex polytope in $(n-3)$-dimensional space, while the Duistermaat-Heckman theorem shows that the pushforward measure on this polytope is Lebesgue measure on $\mathbb{R}^{n-3}$. Together, these theorems allow us to define a measure-preserving set of "action-angle" coordinates on the space of closed equilateral polygons. The new coordinate system allows us to make explicit computations of exact expectations for total curvature and for some chord lengths of closed (and confined) equilateral random walks, to give statistical criteria for sampling algorithms on the space of polygons and to prove that the probability that a randomly chosen equilateral hexagon is unknotted is at least $\frac{1}{2}$. We then use our methods to construct a new Markov chain sampling algorithm for equilateral closed polygons, with a simple modification to sample (rooted) confined equilateral closed polygons. We prove rigorously that our algorithm converges geometrically to the standard measure on the space of closed random walks, give a theory of error estimators for Markov chain Monte Carlo integration using our method and analyze the performance of our method. Our methods also apply to open random walks in certain types of confinement, and in general to walks with arbitrary (fixed) edgelengths as well as equilateral walks.Comment: Published at http://dx.doi.org/10.1214/15-AAP1100 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Statistical and hydrodynamic properties of topological polymers for various graphs showing enhanced short-range correlation

“…We remark that the quaternion method has been generalized to a fast algorithm for generating equilateral random polygons through symplectic geometry. [32,33,34] We analytically show that the ratio of the gyration radius to the hydrodynamic radius of a topological polymer, R G /R H , is characterized by the variance of the probability distribution function of the distance between two segments of the polymer. In particular, we argue that the ratio decreases if the variance becomes small.…”

Random Triangles and Polygons in the Plane

“…In the same ongoing discussion in which Woolhouse posed his versions of the obtuse triangle problem, J. J. Sylvester in 1864 asked for the probability that four points "taken at random in a plane" formed the vertices of a reentrant (embedded, but not convex) quadrilateral [25]. 6 Various solutions were proposed by Cayley [24, footnote 64(b)], De Morgan [7, pp. 147-148], and others, with answers including (at least) 1/4, 35/12π 2 , 3/8, 1/3, and 1/2 [14].…”

Knotting probability of equilateral hexagons

“…In this section, we will discuss the probability that a random equilateral hexagon is knotted. It has been proven that at least 1 3 of hexagons with total length 2 are unknotted [8]. Using action-angle coordinates and Calvo's geometric invariant curl, Cantarella and Shonkwiler [8] prove that at least 1 2 of the space of equilateral hexagons consists of unknots.…”