Jean-Jil Duchamps

ANR project: Graphes Aléatoires pour les Réseaux Phylogénétiques

ANR project ID: ANR-24-CE40-7154

The ANR-funded GARP project (graphes aléatoires pour les réseaux phylogénétiques, i.e. random graphs for phylogenetic network) is active for the 2025-2028 period. It is a starting grant (JCJC), with a main focus on the recruitment of a postdoc in Laboratoire de mathématiques de Besançon, and more precisely within the (informal) research group that is focused on probabilistic modeling of ecology and evolutionary biology. The two-year position was obtained by Frederic Alberti, who starts in fall of 2025.

Works and events supported by the project

• Sharp $L\log L$ Condition for Supercritical Galton-Watson Processes with Countable Types (2025).
  → with M André – arXiv. HAL.
  ShowHide abstract

  We investigate Kesten-Stigum-like results for multi-type Galton-Watson processes with a countable number of types in a general setting, allowing us in particular to consider processes with an infinite total population at each generation. Specifically, a sharp $L\log L$ condition is found under the only assumption that the mean reproduction matrix is positive recurrent in the sense of Vere-Jones (1967). The type distribution is shown to always converge in probability in the recurrent case, and under conditions covering many cases it is shown to converge almost surely.

• The $B_2$ Index of Galled Trees (2024).
  → with F Bienvenu, M Fuchs and TC Yu – arXiv. HAL.
  ShowHide abstract

  In recent years, there has been an effort to extend the classical notion of phylogenetic balance, originally defined in the context of trees, to networks. One of the most natural ways to do this is with the so-called $B_2$ index. In this paper, we study the $B_2$ index for a prominent class of phylogenetic networks: galled trees. We show that the $B_2$ index of a uniform leaf-labeled galled tree converges in distribution as the network becomes large. We characterize the corresponding limiting distribution, and show that its expected value is 2.707911858984... This is the first time that a balance index has been studied to this level of detail for a random phylogenetic network.
  One specificity of this work is that we use two different and independent approaches, each with its advantages: analytic combinatorics, and local limits. The analytic combinatorics approach is more direct, as it relies on standard tools; but it involves slightly more complex calculations. Because it has not previously been used to study such questions, the local limit approach requires developing an extensive framework beforehand; however, this framework is interesting in itself and can be used to tackle other similar problems.