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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A recurrence/transience result
for circle packings


Author: Gareth McCaughan
Journal: Proc. Amer. Math. Soc. 126 (1998), 3647-3656
MSC (1991): Primary 52C15; Secondary 30C35, 30G25, 60J15
MathSciNet review: 1327026
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Abstract: It is known that any infinite simplicial complex homeomorphic to the plane and satisfying a couple of other conditions is the nerve of a circle packing of either the plane or the disc (and not of both). We prove that such a complex is the nerve of a packing of the plane or the disc according as the simple random walk on its 1-skeleton is recurrent or transient, and discuss some applications. We also prove a criterion for transience of simple random walk on the 1-skeleton of a triangulation of the plane, in terms of average degrees of suitable sets of vertices.


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Additional Information

Gareth McCaughan
Affiliation: Department of Pure Mathematics and Mathematical Statistics, Cambridge University, Mill Lane, Cambridge, England
Email: gjm11@pmms.cam.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-98-03353-X
PII: S 0002-9939(98)03353-X
Received by editor(s): August 19, 1994
Received by editor(s) in revised form: February 16, 1995
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1998 American Mathematical Society