Manchester Geometry Seminar 2012/2013

Thursday 13 December 2012. The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm

Probability Theory of Random Polygons from the Quaternionic Viewpoint

Clayton Shonkwiler (University of Georgia)

Here is a natural question in statistical physics: What is the expected shape of a polymer with $n$ monomers in solution? The corresponding mathematical question is equally interesting: Consider the space of $n$-gons in three-dimensional space with length $2$, modulo translations. This is a compact manifold. What is the natural metric (and corresponding probability measure) on this manifold? And what are the statistical properties of $n$-gons in $3$-space sampled uniformly from this probability measure?

In this talk, I will describe a natural probability measure on length $2$ $n$-gon space pushed forward from the standard measure on the Stiefel manifold of $2$-frames in complex $n$-space. The pushforward map comes from a construction of Hausmann and Knutson from algebraic geometry.

We are able to explicitly and exactly compute the expected value of the radius of gyration for polygons sampled from our measure, and also give a fast algorithm for directly sampling the space of closed polygons. The talk describes joint work with Jason Cantarella and Tetsuo Deguchi.