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A recurrence/transience result for circle packings
Author(s):
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.
References:
- 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:
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
Copyright of article:
Copyright
1998,
American Mathematical Society
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