  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.
