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

TL;DR

This paper investigates explosive Crump-Mode-Jagers branching processes in random environments, establishing conditions for infinite offspring at explosion time and applying findings to super-linear preferential attachment models.

Contribution

It weakens assumptions needed to prove the existence of an individual with infinite offspring at explosion and extends analysis to previously unstudied cases in preferential attachment models.

Findings

Proved the existence of a unique individual with infinite offspring at explosion time.

Extended the analysis to a broader class of super-linear preferential attachment models.

Filled gaps in previous case analyses, covering new parameter ranges.

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.