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- W4239432816 abstract "The contacts graph, or nerve, of a packing, is a combinatorial graph that describes the combinatorics of the packing. Let G 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 packing P in the plane (resp. the unit disk) with contacts graph G. Several criteria for deciding whether G 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 on G is recurrent, then G is CP parabolic. Conversely, if G has bounded valence and the random walk on G is transient, then G is CP hyperbolic. We also give a new proof that G 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 if G is CP hyperbolic and D is any simply connected proper subdomain of the plane, then there is a disk packing P with contacts graph G such that P is contained and locally finite in D." @default.
- W4239432816 created "2022-05-12" @default.
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- W4239432816 date "2011-01-01" @default.
- W4239432816 modified "2023-09-25" @default.
- W4239432816 title "Hyperbolic and Parabolic Packings*" @default.
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- W4239432816 doi "https://doi.org/10.1007/978-1-4419-9675-6_7" @default.
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