A recurrence/transience result for circle packings
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- by Gareth McCaughan PDF
- Proc. Amer. Math. Soc. 126 (1998), 3647-3656 Request permission
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.References
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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
- Received by editor(s): August 19, 1994
- Received by editor(s) in revised form: February 16, 1995
- Communicated by: Albert Baernstein II
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 3647-3656
- MSC (1991): Primary 52C15; Secondary 30C35, 30G25, 60J15
- DOI: https://doi.org/10.1090/S0002-9939-98-03353-X
- MathSciNet review: 1327026