  We study homoclinic snaking of one-dimensional, localised states on
two-dimensional, bistable lattices via the method of exponential asymptotics.
Within a narrow region of parameter space, fronts connecting the two stable
states are pinned to the underlying lattice. Localised solutions are formed by
matching two such stationary fronts back-to-back; depending on the orientation
relative to the lattice, the solution branch may `snake' back and forth within
the pinning region via successive saddle-node bifurcations. Standard continuum
approximations in the weakly nonlinear limit (equivalently, the limit of small
mesh size) do not exhibit this behaviour, due to the resultant leading-order
reaction-diffusion equation lacking a periodic spatial structure. By including
exponentially small effects hidden beyond all algebraic orders in the
asymptotic expansion, we find that exponentially small but exponentially
growing terms are switched on via error function smoothing near Stokes lines.
Eliminating these otherwise unbounded beyond-all-orders terms selects the
origin (modulo the mesh size) of the front, and matching two fronts together
yields a set of equations describing the snaking bifurcation diagram. This is
possible only within an exponentially small region of parameter space---the
pinning region. Moreover, by considering fronts orientated at an arbitrary
angle $\psi$ to the $x$-axis, we show that the width of the pinning region is
non-zero only if $\tan\psi$ is rational or infinite. This is the first time a
formula explicitly relating the orientation of a front to the width of its
pinning region has been derived. The asymptotic results are compared with
numerical calculations, with good agreement.
