Ancestral diversity in fragmentation trees
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911324356739072 |
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| author | Haas, Bénédicte Miermont, Grégory |
| author_facet | Haas, Bénédicte Miermont, Grégory |
| contents | In a deterministic or random tree, a notion of ancestral diversity can be defined as follows. Sample independently $n$ groups of $k$ leaves and count the number $N_n(k)$ of distinct most recent common ancestors of each of the groups. As $n$ becomes large, the asymptotic behavior of $N_n(k)$ depends of course on the structure of the tree. Motivated by the study of the edge density in the Brownian co-graphon, Chapuy recently considered this problem in the case where $k=2$ and where the tree is the Brownian continuum random tree. We vastly extend this framework by considering general values of $k$ and general fragmentation trees, which include some prominent examples such as stable Lévy trees and idealized models of phylogenetic trees. Other natural ancestral statistics are also considered. For a given tree model, we identify a phase transition-like phenomenon, with different asymptotic regimes for $N_k(n)$, depending on the position of $k$ relative to a model-dependent critical value. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_15500 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ancestral diversity in fragmentation trees Haas, Bénédicte Miermont, Grégory Probability In a deterministic or random tree, a notion of ancestral diversity can be defined as follows. Sample independently $n$ groups of $k$ leaves and count the number $N_n(k)$ of distinct most recent common ancestors of each of the groups. As $n$ becomes large, the asymptotic behavior of $N_n(k)$ depends of course on the structure of the tree. Motivated by the study of the edge density in the Brownian co-graphon, Chapuy recently considered this problem in the case where $k=2$ and where the tree is the Brownian continuum random tree. We vastly extend this framework by considering general values of $k$ and general fragmentation trees, which include some prominent examples such as stable Lévy trees and idealized models of phylogenetic trees. Other natural ancestral statistics are also considered. For a given tree model, we identify a phase transition-like phenomenon, with different asymptotic regimes for $N_k(n)$, depending on the position of $k$ relative to a model-dependent critical value. |
| title | Ancestral diversity in fragmentation trees |
| topic | Probability |
| url | https://arxiv.org/abs/2512.15500 |