Scaling limits of k-ary growing trees

Bénédicte Haas (1, 2) , Robin Stephenson (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 DMA - Département de Mathématiques et Applications - ENS Paris

For each integer $k \geq 2$, we introduce a sequence of $k$-ary discrete trees constructed recursively by choosing at each step an edge uniformly among the present edges and grafting on ''its middle" $k-1$ new edges. When $k=2$, this corresponds to a well-known algorithm which was first introduced by Rémy. Our main result concerns the asymptotic behavior of these trees as $n$ becomes large: for all $k$, the sequence of $k$-ary trees grows at speed $n^{1/k}$ towards a $k$-ary random real tree that belongs to the family of self-similar fragmentation trees. This convergence is proved with respect to the Gromov-Hausdorff-Prokhorov topology. We also study embeddings of the limiting trees when $k$ varies.

Domaines: Probabilités [math.PR]

Bénédicte Haas, Robin Stephenson. Scaling limits of k-ary growing trees. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2014, pp.Online first. ⟨10.1214/14-AIHP622⟩. ⟨hal-00943049⟩