Densities of short uniform random walks

This article by Jonathan M. Borwein, Armin Straub, James Wan and Wadim Zudilin has been published in Canadian Journal of Mathematics (published online Nov 2011) and is available at doi:10.4153/CJM-2011-079-2.

You can also find this article on the arXiv as 1103.2995 [math.CA].

This paper includes an appendix by Don Zagier.

Abstract

We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and less completely those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed.

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