RANDOM-WALKS AND HARMONIC-FUNCTIONS ON INFINITE PLANAR GRAPHS USING SQUARE TILINGS

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 geo metric boundary of the tiling is a circle and is easy to describe in g eneral. The simple random walk on the graph converges (with probabilit y 1) to a point in the geometric boundary. We obtain information on th e harmonic measure and estimates on the rate of convergence. This allo ws us to extend results we previously proved for triangulations of a d isk.