Talk: Hypergeometric evaluations of the densities of short random walks (AG11)

Date: 2011/10/06
Occasion: SIAM Conference on Applied Algebraic Geometry, Minisymposium on Symbolic Combinatorics
Place: North Carolina State University

Abstract

We consider random walks in the plane which consist of n steps of fixed length each taken into a uniformly random direction. Our interest lies in the probability density function of the distance travelled by such a walk. While Lord Rayleigh's limiting density is an excellent approximation for moderately large n, we seek closed forms for the densities in the case of small n.

One of the goals of the talk will be to show that in the cases n=3 and n=4 hypergeometric evaluations can be given. The basic ingredients are combinatorial properties of the associated even moments, computer algebra as well as a surprising modularity of the densities.

This is joint work with Jonathan M. Borwein, James Wan, and Wadim Zudilin.

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