The geometry of random polygons
Seminar Room 1, Newton Institute
What is the expected shape of a ring polymer in solution? This is a natural question in statistical physics which suggests an equally interesting mathematical question: what are the statistics of the geometric invariants of random, fixed-length n-gons in space? Of course, this requires first answering a more basic question: what is the natural metric (and corresponding probability measure) on the compact manifold of fixed-length n-gons in space modulo translation?
In this talk I will describe a natural metric on this space which is pushed forward from the standard metric on the Stiefel manifold of 2-frames in complex n-space via the coordinatewise Hopf map introduced by Hausmann and Knutson. With respect to the corresponding probability measure it is then possible to prove very precise statements about the statistical geometry of random polygons.
For example, I will show that the expected radius of gyration of an n-gon sampled according to this measure is exactly 1/(2n). I will also demonstrate a simple, linear-time algorithm for directly sampling polygons from this measure. This is joint work with Jason Cantarella (University of Georgia, USA) and Tetsuo Deguchi (Ochanomizu University, Japan).