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Explosion and linear transit times in infinite trees

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

$$\begin{aligned} \lim _{n\rightarrow \infty }m_n(T)\, <\, \infty \,, \end{aligned}$$

and say that T exhibits linear growth if

$$\begin{aligned} \liminf _{n\rightarrow \infty }\, \frac{m_n(T)}{n}\, > \, 0\,. \end{aligned}$$

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.

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Notes

  1. 1.

    We use the standard definition \(G(x)=\mathbb {P}(X\le x)\) for the distribution function.

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Acknowledgments

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.

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Correspondence to Omid Amini.

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Amini, O., Devroye, L., Griffiths, S. et al. Explosion and linear transit times in infinite trees. Probab. Theory Relat. Fields 167, 325–347 (2017). https://doi.org/10.1007/s00440-015-0683-z

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Mathematics Subject Classification

  • 60K35
  • 60C05
  • 60G55
  • 60J80
  • 05C80