A star is born: Explosive Crump-Mode-Jagers branching processes
Bas Lodewijks
TL;DR
We analyze explosive CMJ branching processes in random environment and prove that explosion yields a unique individual with infinite offspring at the explosion time, under weakened assumptions relative to prior work. The approach hinges on a general set of sufficient criteria involving dominating random variables $Y_n$ and a diverging sequence $\lambda_n$, with a key summability condition $\sum_n \mathcal{M}_{\lambda_n}(Y_n)\mathcal{L}_{\lambda_n}(\mathcal{P}_n(\varnothing))<\infty$. The results are then applied to super-linear preferential attachment trees with fitness, linking the CMJ framework to growth dynamics and condensation phenomena, and yielding conditions under which the limiting tree $T_\infty$ contains a unique infinite-star node and no infinite path. The paper extends prior work (e.g., IyerLod23, Iyer24) to a broader class of fitness-growth regimes, providing a broad toolkit for analyzing explosion-induced condensation in random-environment branching processes and their graph-theoretic counterparts.
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.