Hyperbolic and parabolic packings

The contacts graph, or nerve, of a packing, is a combinatorial graph that describes the combinatorics of the packing. LetG be the 1-skeleton of a triangulation of an open disk.G is said to be CP parabolic (resp. CP hyperbolic) if there is a locally finite disk packingP in the plane (resp. the unit disk) with contacts graphG. Several criteria for deciding whetherG is CP parabolic or CP hyperbolic are given, including a necessary and sufficient combinatorial criterion. A criterion in terms of the random walk says that if the random walk onG is recurrent, theG is CP parabolic. Conversely, ifG has bounded valence and the random walk onG is transient, thenG is CP hyperbolic. We also give a new proof thatG is either CP parabolic or CP hyperbolic, but not both. The new proof has the advantage of being applicable to packings of more general shapes. Another new result is that ifG is CP hyperbolic andD is any simply connected proper subdomain of the plane, then there is a disk packingP with contacts graphG such thatP is contained and locally finite inD. Both authors acknowledge support by NSF grants. The first author was also supported by the A. Sloan Research Fellowship.