The geometry and topology of random polygons
Seminar Room 2, Newton Institute Gatehouse
AbstractHere 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 translation. 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, we 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 will be 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 Malcolm Adams (University of Georgia, USA), Tetsuo Deguchi (Ochanomizu University, Japan), and Clay Shonkwiler (University of Georgia, USA).
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