Condensation in subcritical Cauchy Bienaymé trees

TL;DR

This work resolves the height and condensation behavior of large subcritical Bienaymé trees whose offspring tails fall in the Cauchy domain ($\alpha=1$). By extending the one-big-jump principle to the $\alpha=1$ regime and leveraging local estimates, the authors prove a condensation phenomenon with a unique vertex of macroscopic degree and show the tree height grows logarithmically with size, $\mathsf{H}(\mathcal{T}_n) / \log n \to_p 1/\log(1/m)$. They also establish precise limits for the condensation vertex height and quantify when the height fluctuations are tight, tying tightness to the finiteness of $\sum_{n\ge1} n\log n \mu_n$, and discuss how the $\alpha=1$ case mirrors $\alpha \in (1,2]$ rather than the critical $\alpha=1$ case. The results complete the height analysis for heavy-tailed size-conditioned Bienaymé trees and connect to a broad range of combinatorial structures via the one-big-jump mechanism.

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 $α>1$.

Figures (1)

• Figure 1: A simulation of a subcritical Cauchy Bienaymé tree with $20000$ vertices.

Theorems & Definitions (16)

• theorem 1
• proposition 1: Kennedy
• Lemma 2
• proof
• Example 3
• theorem 4
• proof
• Corollary 5
• proposition 2
• proposition 3