“Probability theory of random polygons from the quaternionic perspective”
Communications on Pure and Applied Mathematics (2013), to appear.
Joint with Jason Cantarella and Tetsuo Deguchi.
arXiv: 1206.3161 [math.DG]
In this paper we defined a new measure on the space of polygons of fixed length by pushing forward Haar measure on the complex Stiefel manifold \(V_2(\mathbb{C}^n)\) of 2-frames in \(\mathbb{C}^n\) to the space of fixed-length polygons in \(\mathbb{R}^3\) using the coordinatewise Hopf map described by Hausmann and Knutson. The advantage of this measure, which we call the symmetric measure, is that we can use the Riemannian geometry of the Stiefel manifold to understand the probability theory of polygons.
For example, we showed that the expected radius of gyration of a polygon in the symmetric measure is exactly \(\frac{1}{2n}\).1 Moreover, sampling with respect to this measure is simple to code and linear in the number of edges, so it is straightforward—if we want to test some hypothesis—to produce a few million polygons and see what the answer is on that sample. We can produce polygons with very large numbers of edges almost instantaneously; the animation above shows a random 20,000-gon, and here’s a random 1,000,000-gon (click to see full size):
Please see the paper for many more details.
-
You can see the definition at the above-linked Wikipedia article, but the radius of gyration of any shape is some sort of average distance of the points in the object from its center of mass. In the case of polymers, it's a useful measure of size since it can actually be determined experimentally. ↩