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 ${\Bbb R}^2$ whose contacts graph is a bounded valence triangulation of a disk, then, with probability $1$ , the simple random walk on $P$ converges to a limit point. Moreover, in this situation any continuous function on the limit set 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 that 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 isometric to a planar graph.