Proceedings of the American Mathematical Society
It is shown there that an infinite connected planar graph with a uniform upper bound on vertex degree and rapidly decreasing Green's function (relative to the simple random walk) has infinitely many pairwise finitely-intersecting geodesic rays starting at each vertex. We then demonstrate the existence of nonconstant bounded harmonic functions on the graph.
Northshield, S. (1991). Geodesics and Bounded Harmonic Functions on Infinite Graphs. Proceedings of the American Mathematical Society, 113 (1), 229-233.