## Abstract

Let T be an infinite rooted tree with weights w_{e} assigned to its edges. Denote by m_{n}(T) the minimum weight of a path from the root to a node of the nth generation. We consider the possible behaviour of m_{n}(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 language | English |
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Pages (from-to) | 325-347 |

Number of pages | 23 |

Journal | Probability Theory and Related Fields |

Volume | 167 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2017 |

### Funding

This project grew out of discussions started at the McGill Universityâs Bellairs Institute, Barbados. We would like to thank Nicolas Curien for asking a question which led to the results in Sect.Â 5 , and thank BÃ©nÃ©dicte 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 LemmaÂ 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.

Funders | Funder number |
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Engineering and Physical Sciences Research Council | EP/J019496/1 |

Nederlandse Organisatie voor Wetenschappelijk Onderzoek |