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

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. [ABSTRACT FROM AUTHOR]