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Area Distribution of Two-Dimensional Random Walks on a Square Lattice

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Abstract

The algebraic area probability distribution of closed planar random walks of length N on a square lattice is considered. The generating function for the distribution satisfies a recurrence relation in which the combinatorics is encoded. A particular case generalizes the q-binomial theorem to the case of three addends. The distribution fits the Lévy probability distribution for Brownian curves with its first-order 1/N correction quite well, even for N rather small.

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Authors and Affiliations

  1. Schrödinger, 120 West 45th St., New York, NY, 10036, USA

    Stefan Mashkevich

  2. Bogolyubov Institute for Theoretical Physics, 03143, Kiev, Ukraine

    Stefan Mashkevich

  3. Laboratoire de Physique Théorique et Modèles Statistiques, CNRS-Paris Sud, UMR 8626, Université Paris-Sud, 91405, Orsay, France

    Stéphane Ouvry

Authors
  1. Stefan Mashkevich
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  2. Stéphane Ouvry
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Correspondence to Stefan Mashkevich.

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Mashkevich, S., Ouvry, S. Area Distribution of Two-Dimensional Random Walks on a Square Lattice. J Stat Phys 137, 71–78 (2009). https://doi.org/10.1007/s10955-009-9827-9

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  • DOI: https://doi.org/10.1007/s10955-009-9827-9

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