  A variety of transport processes in natural and man-made systems are
intrinsically random. To model their stochasticity, lattice random walks have
been employed for a long time, mainly by considering Cartesian lattices.
However, in many applications in bounded space the geometry of the domain may
have profound effects on the dynamics and ought to be accounted for. We
consider here the cases of the six-neighbour (hexagonal) and three-neighbour
(honeycomb) lattice, which are utilised in models ranging from adatoms
diffusing in metals and excitations diffusing on single-walled carbon nanotubes
to animal foraging strategy and the formation of territories in scent-marking
organisms. In these and other examples, the main theoretical tool to study the
dynamics of lattice random walks in hexagonal geometries has been via
simulations. Analytic representations have in most cases been inaccessible, in
particular in bounded hexagons, given the complicated zig-zag boundary
conditions that a walker is subject to. Here we generalise the method of images
to hexagonal geometries and obtain closed-form expressions for the occupation
probability, the so-called propagator, for lattice random walks both on
hexagonal and honeycomb lattices with periodic, reflective and absorbing
boundary conditions. In the periodic case, we identify two possible choices of
image placement and their corresponding propagators. Using them, we construct
the exact propagators for the other boundary conditions, and we derive
transport related statistical quantities such as first passage probabilities to
one or multiple targets and their means, elucidating the effect of the boundary
condition on transport properties.
