Spinal partitions and invariance under re-rooting of continuum random trees

Bénédicte Haas (1) , Jim Pitman (2) , Matthias Winkel (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Department of Statistics [Berkeley]
3 Department of Statistics [Oxford]

We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a two-parameter Poisson-Dirichlet family of continuous fragmentation trees, including the stable trees of Duquesne and Le Gall, the fine partition is obtained from the coarse one by shattering each of its parts independently, according to the same law. As a second application of spinal decompositions, we prove that among the continuous fragmentation trees, stable trees are the only ones whose distribution is invariant under uniform re-rooting.

Mots clés: Markov branching model; discrete tree; Poisson-Dirichlet distribution; fragmentation process; continuum random tree; spinal decomposition; random re-rooting

Domaines: Probabilités [math.PR]

Bénédicte Haas, Jim Pitman, Matthias Winkel. Spinal partitions and invariance under re-rooting of continuum random trees. Annals of Probability, 2009, 37 (4), pp.1381-1411. ⟨hal-00149050⟩