Condensation in subcritical Cauchy Bienaymé trees
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908581557698560 |
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| author | Kortchemski, Igor Vetter, Leonard |
| author_facet | Kortchemski, Igor Vetter, Leonard |
| contents | 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 $α>1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07530 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Condensation in subcritical Cauchy Bienaymé trees Kortchemski, Igor Vetter, Leonard Probability Primary 60J80, 60G50, 05C05, (Secondary) 05C80 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 $α>1$. |
| title | Condensation in subcritical Cauchy Bienaymé trees |
| topic | Probability Primary 60J80, 60G50, 05C05, (Secondary) 05C80 |
| url | https://arxiv.org/abs/2503.07530 |