2025/03/10 by Kortchemski, Igor, Vetter, Leonard
#(Secondary) 05C80 #05C05 #60G50 #FOS: Mathematics #Primary 60J80 #Probability (math.PR)
paper · doi:10.48550/arxiv.2503.07530
The goal of this note is to study the geometry of large size-conditioned Bienaymé trees whose offspring distribution is subcritical, belongs to the domain of attraction of a stable law of index α=1 and satisfies a local regularity assumption. We show that a condensation phenomenon occurs: one unique vertex of macroscopic degree emerges, and its height converges in distribution to a geometric random variable. Furthermore, the height of such trees grows logarithmically in their size. Interestingly, the behavior of subcritical Bienaymée trees with α=1 is quite similar to the case α∈( 1,2], in contrast with the critical case. This completes the study of the height of heavy-tailed size-conditioned Bienaymé trees. Our approach is to check that a random-walk one-big-jump principle due to Armendáriz & Loulakis holds, by using local estimates due to Berger, combined with the previous approach to study subcritical Bienaymé trees with α>1.