A recurrence/transience result for circle packings

McCaughan, Gareth

doi:10.1090/S0002-9939-98-03353-X

A recurrence/transience result for circle packings
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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.

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