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

TL;DR

This paper analyzes the algebraic area distribution of closed 2D random walks on a square lattice, deriving a generating function that generalizes the q-binomial theorem and showing it aligns with Levy distribution for Brownian curves.

Contribution

It introduces a recurrence relation for the generating function of the algebraic area distribution, generalizes the q-binomial theorem, and connects the distribution to Levy distribution with finite-size corrections.

Findings

Distribution fits Levy distribution well

Recurrence relation encodes combinatorics

Generalization of q-binomial theorem

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 L\'evy probability distribution for Brownian curves with its first-order 1/N correction quite well, even for N rather small.