Condensation in subcritical Cauchy Bienaymé trees

Abstract

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 $α\in( 1,2]$, 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 $α>1$.