Scaling limits of Markov branching trees, with applications to Galton-Watson and random unordered trees

Benedicte Haas (1) , Grégory Miermont (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We consider a family of random trees satisfying a Markov branching property. Roughly, this property says that the subtrees above some given height are independent with a law that depends only on their total size, the latter being either the number of leaves or vertices. Such families are parameterized by sequences of distributions on partitions of the integers, that determine how the size of a tree is distributed in its different subtrees. Under some natural assumption on these distributions, stipulating that ``macroscopic'' splitting events are rare, we show that Markov branching trees admit the so-called self-similar fragmentation trees as scaling limits in the Gromov-Hausdorff-Prokhorov topology. Applications include scaling limits of consistent Markov branching model, and convergence of Galton-Watson trees towards the Brownian and stable continuum random trees. We also obtain that random uniform unordered trees have the Brownian tree as a scaling limit, hence extending a result by Marckert-Miermont and fully proving a conjecture made by Aldous.

Mots clés: Markov branching trees; scaling limits; fragmentation trees; Brownian tree; stable trees; random unordered trees

Domaines: Probabilités [math.PR]

Benedicte Haas, Grégory Miermont. Scaling limits of Markov branching trees, with applications to Galton-Watson and random unordered trees. Annals of Probability, 2012, 40, pp.2589-2666. ⟨hal-00464337⟩