Explosion and linear transit times in infinite trees

O. Amini, L. Devroye, S. Griffiths, N.K. Olver

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

Let T be an infinite rooted tree with weights we assigned to its edges. Denote by mn(T) the minimum weight of a path from the root to a node of the nth generation. We consider the possible behaviour of mn(T) with focus on the two following cases: we say T is explosive if limn→∞mn(T)<∞,and say that T exhibits linear growth if lim infn→∞mn(T)n>0.We consider a class of infinite randomly weighted trees related to the Poisson-weighted infinite tree, and determine precisely which trees in this class have linear growth almost surely. We then apply this characterization to obtain new results concerning the event of explosion in infinite randomly weighted spherically-symmetric trees, answering a question of Pemantle and Peres (Ann Probab 22(1), 180–194, 1994). As a further application, we consider the random real tree generated by attaching sticks of deterministic decreasing lengths, and determine for which sequences of lengths the tree has finite height almost surely.

Original languageEnglish
Pages (from-to)325-347
Number of pages23
JournalProbability Theory and Related Fields
Volume167
Issue number1-2
DOIs
Publication statusPublished - 2017

Funding

This project grew out of discussions started at the McGill University’s Bellairs Institute, Barbados. We would like to thank Nicolas Curien for asking a question which led to the results in Sect. 5 , and thank Bénédicte Haas and him for sharing a draft of their work at the final stage of the preparation of this paper. We thank the anonymous referee for a thorough reading and pointing out the shorter proof of Lemma 2.2 . Special thanks to the Brazilian-French Network in Mathematics for providing generous support for a visit of S.G. at ENS Paris. S.G. was supported by EPSRC grant EP/J019496/1. N.O. was supported by a NWO Veni grant. This project grew out of discussions started at the McGill University’s Bellairs Institute, Barbados. We would like to thank Nicolas Curien for asking a question which led to the results in Sect. , and thank Bénédicte Haas and him for sharing a draft of their work at the final stage of the preparation of this paper. We thank the anonymous referee for a thorough reading and pointing out the shorter proof of Lemma . Special thanks to the Brazilian-French Network in Mathematics for providing generous support for a visit of S.G. at ENS Paris. S.G. was supported by EPSRC grant EP/J019496/1. N.O. was supported by a NWO Veni grant.

FundersFunder number
Engineering and Physical Sciences Research CouncilEP/J019496/1
Nederlandse Organisatie voor Wetenschappelijk Onderzoek

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