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In this paper we investigate the long-term behaviour of solutions to thediscrete Allen-Cahn equation posed on a two-dimensional lattice. We show thatfront-like initial conditions evolve towards a planar travelling wave modulatedby a phaseshift $\gamma_l(t)$ that depends on the coordinate $l$ transverse tothe primary direction of propagation. This direction is allowed to be general,but rational, generalizing earlier known results for the horizontal direction.We show that the behaviour of $\gamma$ can be asymptotically linked to thebehaviour of a suitably discretized mean curvature flow. This allows us to showthat travelling waves propagating in rational directions are nonlinearly stablewith respect to perturbations that are asymptotically periodic in thetransverse direction.

Curvature-driven front propagation through planar lattices in oblique directions