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A counterexample to the conjecture of Woess on simple random walks on trees

Authors: Kenneth A. Berman and Mokhtar Konsowa
Journal: Proc. Amer. Math. Soc. 105 (1989), 443-449
MSC: Primary 60J15; Secondary 05C05
MathSciNet review: 936772
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Abstract: Let $ T$ be a locally finite tree with a countable number of vertices. The volume of $ T$ is the energy dissipation of the unit flow from the root of infinity that divides equally at every branching of the tree. It follows from Thomson's Principle that if $ T$ contains an infinite leafless subtree whose volume is finite then $ T$ is transient. Woess [6] conjectured that the converse is also true. In this paper we give a counterexample to this conjecture by constructing a transient tree, such that every infinite leafless subtree has infinite volume.

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Article copyright: © Copyright 1989 American Mathematical Society