Jonathan M. Borwein ; Dirk Nuyens ; Armin Straub ; James Wan
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Random Walks in the Plane
dmtcs:2862 -
Discrete Mathematics & Theoretical Computer Science,
January 1, 2010,
DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)
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https://doi.org/10.46298/dmtcs.2862
Random Walks in the Plane
Authors: Jonathan M. Borwein ; Dirk Nuyens ; Armin Straub ; James Wan
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Jonathan M. Borwein;Dirk Nuyens;Armin Straub;James Wan
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.