We study a wide class of transient planar graphs, through a geometric model given by a square tiling of a cylinder. For many graphs, the geometric boundary of the tiling is a circle and is easy to describe in general. The simple random walk on the graph converges (with probability 1) to a point in the geometric boundary. We obtain information on the harmonic measure and estimates on the rate of convergence. This allows us to extend results we previously proved for triangulations of a disk.
"Random walks and harmonic functions on infinite planar graphs using square tilings." Ann. Probab. 24 (3) 1219 - 1238, July 1996. https://doi.org/10.1214/aop/1065725179