Analysis of random walks on a hexagonal lattice

TL;DR

This paper analyzes a discrete-time random walk on a hexagonal lattice, deriving probability functions, moments, and asymptotic behavior, and explores first-passage-time probabilities with potential applications.

Contribution

It provides a comprehensive analysis of random walks on hexagonal lattices, including convergence to Brownian motion and explicit first-passage-time formulas, which are novel contributions.

Findings

Derived probability generating functions and transition probabilities.

Established convergence to 2D Brownian motion.

Obtained closed-form first-passage-time probabilities.

Abstract

We consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice. We determine the probability generating functions, the transition probabilities and the relevant moments. The convergence of the stochastic process to a 2-dimensional Brownian motion is also discussed. Furthermore, we obtain some results on its asymptotic behavior making use of large deviation theory. Finally, we investigate the first-passage-time problem of the random walk through a vertical straight-line. Under suitable symmetry assumptions we are able to determine the first-passage-time probabilities in a closed form, which deserve interest in applied fields.