Probability Theory of Random Polygons from the Quaternionic Viewpoint

Abstract

We build a new probability measure on closed space and plane polygons. The key construction is a map, given by Knutson and Hausmann using the Hopf map on quaternions, from the complex Stiefel manifold of 2-frames in n-space to the space of closed n-gons in 3-space of total length 2. Our probability measure on polygon space is defined by pushing forward Haar measure on the Stiefel manifold by this map. A similar construction yields a probability measure on plane polygons which comes from a real Stiefel manifold. The edgelengths of polygons sampled according to our measures obey beta distributions. This makes our polygon measures different from those usually studied, which have Gaussian or fixed edgelengths. One advantage of our measures is that we can explicitly compute expectations and moments for chordlengths and radii of gyration. Another is that direct sampling according to our measures is fast (linear in the number of edges) and easy to code. Some of our methods will be of independent interest in studying other probability measures on polygon spaces. We define an edge set ensemble (ESE) to be the set of polygons created by rearranging a given set of n edges. A key theorem gives a formula for the average over an ESE of the squared lengths of chords skipping k vertices in terms of k, n, and the edgelengths of the ensemble. This allows one to easily compute expected values of squared chordlengths and radii of gyration for any probability measure on polygon space invariant under rearrangements of edges.

Some small typos fixed, added a calculation for the covariance of edgelengths, added pseudocode for the random polygon sampling algorithm. To appear in Communications on Pure and Applied Mathematics (CPAM)

References in corpus (4)

• On the Limits of Analogy Between Self-Avoidance and Topology-Driven Swelling of Polymer Loops
• The Expected Total Curvature of Random Polygons
• Characteristics of shape and knotting in ideal rings
• The Geometry and Topology on Grassmann Manifolds

Cited by in corpus (11)

• Statistical and hydrodynamic properties of topological polymers for various graphs showing enhanced short-range correlation
• The symplectic geometry of closed equilateral random walks in 3-space
• Asymptotic laws for random knot diagrams
• The Expected Total Curvature of Random Polygons
• Random Triangles and Polygons in the Plane
• Admissibility and Frame Homotopy for Quaternionic Frames
• Knot probabilities in equilateral random polygons
• Proteins analysed as virtual knots
• Efficient Mesh Optimization Using the Gradient Flow of the Mean Volume
• Stiefel manifolds and polygons
• Spherical Geometry and the Least Symmetric Triangle