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

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.

8 pages, LaTeX 2e. Reformulated in terms of q-commutators

References in corpus (2)

• Exact Random Walk Distributions using Noncommutative Geometry
• Distribution of the area enclosed by a 2D random walk in a disordered medium

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