2025/12/17 by Haas, Bénédicte, Miermont, Grégory
Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · doi:10.48550/arxiv.2512.15500
openalex publication_date 2025/12/17 · openalex created_date 2025/12/19 · openalex updated_date 2026/07/28
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 Nn(k) of distinct most recent common ancestors of each of the groups. As n becomes large, the asymptotic behavior of Nn(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 Nk(n), depending on the position of k relative to a model-dependent critical value.