Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic mod\-els

DOI10.1214/07-AOP377zbMath1155.92033arXivmath/0604350OpenAlexW4299129844MaRDI QIDQ948746

Grégory Miermont, Matthias Winkel, Bénédicte Haas, Jim W. Pitman

Publication date: 20 October 2008

Published in: The Annals of Probability (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0604350

zbMATH Keywords

phylogenetic treescontinuum random treeself-similar fragmentationMarkov branching model\(\mathbb R\)-tree

Mathematics Subject Classification ID

Problems related to evolution (92D15) Applications of branching processes (60J85) Branching processes (Galton-Watson, birth-and-death, etc.) (60J80)

Related Items (34)

Bivariate Markov chains converging to Lamperti transform Markov additive processes ⋮ A representation of exchangeable hierarchies by sampling from random real trees ⋮ Gibbs distributions for random partitions generated by a fragmentation process ⋮ Spinal partitions and invariance under re-rooting of continuum random trees ⋮ Exchangeable and sampling-consistent distributions on rooted binary trees ⋮ Restricted exchangeable partitions and embedding of associated hierarchies in continuum random trees ⋮ Trees within trees. II: Nested fragmentations ⋮ Schröder’s problems and scaling limits of random trees ⋮ Self-similar growth fragmentations as scaling limits of Markov branching processes ⋮ The stable trees are nested ⋮ Fires on trees ⋮ Regenerative tree growth: Markovian embedding of fragmenters, bifurcators, and bead splitting processes ⋮ Distributions of linear functionals of two parameter Poisson-Dirichlet random measures ⋮ Growing random graphs with a preferential attachment structure ⋮ Stable graphs: distributions and line-breaking construction ⋮ A binary embedding of the stable line-breaking construction ⋮ Scaling Limits of Markov-Branching Trees and Applications ⋮ Scaling limits of \(k\)-ary growing trees ⋮ Tail asymptotics for extinction times of self-similar fragmentations ⋮ Random tree growth by vertex splitting ⋮ Markov branching in the vertex splitting model ⋮ ENTROPY AND HAUSDORFF DIMENSION IN RANDOM GROWING TREES ⋮ Scaling limits of Markov branching trees with applications to Galton-Watson and random unordered trees ⋮ Distributional properties of means of random probability measures ⋮ Lamperti-type laws ⋮ Gibbs fragmentation trees ⋮ Scaling limits of multi-type Markov branching trees ⋮ A phase transition for the heights of a fragmentation tree ⋮ Generalized Markov branching trees ⋮ Regenerative tree growth: binary self-similar continuum random trees and Poisson-Dirichlet compositions ⋮ Spaces of algebraic measure trees and triangulations of the circle ⋮ Gibbs partitions, Riemann–Liouville fractional operators, Mittag–Leffler functions, and fragmentations derived from stable subordinators ⋮ Fragmentations with self-similar branching speeds ⋮ Scaling limits of Markov branching trees and Galton-Watson trees conditioned on the number of vertices with out-degree in a given set

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