Area distribution of two-dimensional random walks on a square lattice

Stefan Mashkevich 1, 2, Stéphane Ouvry 3

Journal of Statistical Physics 137 (2009) 71-78

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.

  • 1. Schrödinger,
    Schrodinger
  • 2. Bogolyubov Institute for Theoretical Physics,
    Bogolyobov Institute for Theoretical Physics
  • 3. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud