Discrete & Computational Geometry

, Volume 14, Issue 2, pp 123–149 | Cite as

Hyperbolic and parabolic packings

  • Zheng-Xu He
  • O. Schramm


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.


Accumulation Point Jordan Curve Boundary Vertex Circle Packing Extremal Length 
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Copyright information

© Springer-Verlag New York Inc. 1995

Authors and Affiliations

  • Zheng-Xu He
    • 1
  • O. Schramm
    • 2
  1. 1.University of California at San DiegoLa JollaUSA
  2. 2.Mathematics DepartmentThe Weizmann InstituteRehovotIsrael

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