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

DOI:
https://doi.org/10.1090/S0002-9939-98-03353-X

MathSciNet review:
1327026

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Abstract | References | Similar Articles | Additional Information

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.

**1.**A. Ancona,*Théorie du potentiel sur les graphes et les varietés*, Ecole d'Eté de Probabilités de Saint-Flour XVIII, 1988, pp. 5-116.**2.**A. F. Beardon & K. Stephenson,*The uniformization theorem for circle packings*, Indiana U. Math. J.**39**(1990), 1383-1425. MR**92b:52038****3.**J. Dodziuk,*Difference equations, isoperimetric inequality and transience of certain random walks*, Trans. American Math. Soc.**284**(1984), 787-794. MR**85m:58185****4.**G. J. MCaughan,*Some results on circle packings*, Ph.D. thesis, University of Cambridge, 1996.**5.**B. Rodin & D. Sullivan,*The convergence of circle packings to the Riemann mapping*, J. Differential Geom.**26**(1987), 349-360. MR**90c:30007****6.**C. Thomassen,*Isoperimetric inequalities and transient random walks on graphs*, Ann. Prob.**20**(1992), 1592-1600. MR**94a:60106**

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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:
https://doi.org/10.1090/S0002-9939-98-03353-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