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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.


References [Enhancements On Off] (What's this?)

  • [1] Kai Lai Chung, Markov chains with stationary transition probabilities, Second edition. Die Grundlehren der mathematischen Wissenschaften, Band 104, Springer-Verlag New York, Inc., New York, 1967. MR 0217872
  • [2] Peter G. Doyle and J. Laurie Snell, Random walks and electric networks, Carus Mathematical Monographs, vol. 22, Mathematical Association of America, Washington, DC, 1984. MR 920811
  • [3] P. Gerl, Rekurrente und transient Baume, Seminaire Lotharingien de Combinatoire 10, IRMA, Strasbourg, 1984, pp. 80-87.
  • [4] Terry Lyons, A simple criterion for transience of a reversible Markov chain, Ann. Probab. 11 (1983), no. 2, 393–402. MR 690136
  • [5] C. St. J. A. Nash-Williams, Random walk and electric currents in networks, Proc. Cambridge Philos. Soc. 55 (1959), 181–194. MR 0124932
  • [6] Wolfgang Woess, Transience and volumes of trees, Arch. Math. (Basel) 46 (1986), no. 2, 184–192. MR 834834, 10.1007/BF01197498

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DOI: https://doi.org/10.1090/S0002-9939-1989-0936772-2
Article copyright: © Copyright 1989 American Mathematical Society