Talk: An application of modular forms to short random walks (Aachen)

Date: 2012/08/10
Occasion: 1st EU-US Conference on Automorphic Forms and Related Topics
Place: RWTH Aachen, DE

This presentation, given at the 2012 Aachen: Building Bridges conference, gives an overview over some of the results of Densities of short uniform random walks (with an appendix by Don Zagier).

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 distribution of the distance travelled by such a walk. While excellent asymptotic expressions are known for the density functions when n is moderately large, we focus on the arithmetic properties of short random walks.

In the case of three and four steps, the density functions satisfy differential equations of modular origin. This intertwines with the combinatorics of the corresponding even moments and leads to hypergeometric evaluations of the density functions. Much less is known for the density in case of five random steps, but we use the modularity of the four-step case and the Chowla-Selberg formula to deduce its exact behaviour near zero.

The talk will be based on joint work with Jonathan M. Borwein, James Wan, and Wadim Zudilin.

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