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arXiv · 2411.18749

A star is born: Explosive Crump-Mode-Jagers branching processes

Bas Lodewijks
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Abstract

We study a family of Crump--Mode--Jagers branching processes in random environment that explode, i.e. that grow infinitely large in finite time with positive probability. Building on recent work of the author and Iyer (``On the structure of genealogical trees associated with explosive Crump--Mode--Jagers branching processes", arXiv:2311.14664, 2023), we weaken certain assumptions required to prove that the branching process, at the time of explosion, contains a (unique) individual with infinite offspring. We then apply these results to super-linear preferential attachment models. In particular, we fill gaps in some of the cases analysed in Appendix A of the work of the author and Iyer and study a large range of previously unattainable cases.

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Bas Lodewijks. 2024-11-27. A star is born: Explosive Crump-Mode-Jagers branching processes. https://arxiv.org/abs/2411.18749

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