Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models

By Bénédicte Haas, Grégory Miermont, Jim Pitman, Matthias Winkel. 2006

90 citations recorded by the source corpus.

Given any regularly varying dislocation measure, we identify a natural self-similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous's beta-splitting models and Ford's alpha models for phylogenetic trees. This confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf.

Subfields: Mathematical Physics, Molecular Biology, Paleontology

View Paper: https://www.semanticscholar.org/paper/0e38e6bd9afb9f3df6dee2cb098d794282170094

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Citations in the corpus

A new family of Markov branching trees: the alpha-gamma model (2008, methodology): “Lemma 15 (Corollary 22 in [16]).”
Scaling limits of Markov branching trees and Galton-Watson trees conditioned on the number of vertices with out-degree in a given set (2011, background): “Much of the focus has been on limits of trees satisfying either nice consistency relations between trees of various sizes ([8]) or having a nice encoding as continuous functions on [0,∞) (see [9] for an overview).”
Spinal partitions and invariance under re-rooting of continuum random trees (2007, methodology): “In a setting of discrete time and partitions of [n], we postulate that each nonsingleton block splits at each time, which leads to Markov branching models [4, 18, 26].”
Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions (2008, unknown): “COMPOSITIONS1 By Jim Pitman and Matthias Winkel”
Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees (2010, background): “3 Consistent Markov branching models Considering again in a more specific way the Markov branching models, we stress that Theorem 1 also encompasses the results of [25], which hold for particular families (qn, n ≥ 1) satisfying a further consistency property.”