Random Walks in the Plane

This article by Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan has been published in Discrete Mathematics and Theoretical Computer Science (Special volume for FPSAC 2010).

Abstract

We study the expected distance of a two-dimensional walk in the plane with unit steps in random directions. A series evaluation and recursions are obtained making it possible to explicitly formulate this distance for small number of steps. Formulae for all the moments of a 2-step and a 3-step walk are given, and an expression is conjectured for the 4-step walk. The paper makes use of the combinatorical features exhibited by the even moments which, for instance, lead to analytic continuations of the underlying integral.

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