HARMONIC-FUNCTIONS ON PLANAR AND ALMOST PLANAR GRAPHS AND MANIFOLDS, VIA CIRCLE PACKINGS

The circle packing theorem is used to show that on any bounded valence transient planar graph there exists a non constant, harmonic, bounded , Dirichlet function. If P is a bounded circle packing in IR(2) whose contacts graph is a bounded valence triangulation of a disk, then, wit h probability 1, the simple random walk on P converges to a limit poin t. Moreover, in this situation any continuous function on the limit se t of P extends to a continuous harmonic function on the closure of the contacts graph of P; that is, this Dirichlet problem is solvable. We define the notions of almost planar graphs and manifolds, and show tha t under the assumptions of transience and bounded local geometry these possess non constant, harmonic, bounded, Dirichlet functions. Let us stress that an almost planar graph is not necessarily roughly isometri c to a planar graph.