Analysis of random walks on a hexagonal lattice

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 two-dimensional Brownian motion is also discussed. Furthermore, we obtain some results on its asymptotic behaviour 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.