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

Berman, Kenneth A.; Konsowa, Mokhtar

doi:10.1090/S0002-9939-1989-0936772-2

A counterexample to the conjecture of Woess on simple random walks on trees
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by Kenneth A. Berman and Mokhtar Konsowa PDF

Proc. Amer. Math. Soc. 105 (1989), 443-449 Request permission

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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