Limit of trees with fixed degree sequence

Arthur Blanc-Renaudie (1)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

We use a new stick-breaking construction, also introduced independently by Addario-Berry, Donderwinkel, Maazoun, and Martin in arXiv:2107.09726, to study uniform rooted trees with fixed degree sequence D (D-trees). This construction allows us to prove, under natural conditions of convergence of the degree sequence, that D-trees converge toward either P-trees or ICRT. Alongside we provide an upper-bound for the height of D-trees. We deduce similar results concerning P-trees and ICRT. In passing we confirm a conjecture of Aldous, Miermont, and Pitman arXiv:math/0401115 stating that Lévy trees are ICRT with random parameters.

Domaines: Probabilités [math.PR]

Arthur Blanc-Renaudie. Limit of trees with fixed degree sequence. 2021. ⟨hal-03872830⟩