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

  • Stefan Mashkevich
  • Stéphane Ouvry


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.


Random walks q-binomials 


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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.SchrödingerNew YorkUSA
  2. 2.Bogolyubov Institute for Theoretical PhysicsKievUkraine
  3. 3.Laboratoire de Physique Théorique et Modèles Statistiques, CNRS-Paris Sud, UMR 8626Université Paris-SudOrsayFrance

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