Occasion: AMS Fall Western Sectional Meeting 2012, Special Session on Harmonic Maass Forms and q-Series
Place: University of Arizona
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.
|2012densities-tucson.pdf||1.11 MB||Slides (PDF, 74 pages)||988|