Scaling limits of k-ary growing trees
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 4, pp. 1314-1341.

For each integer k2, 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 the number of steps n of the algorithm 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.

Pour chaque entier k2, on introduit une suite d’arbres discrets k-aires construite récursivement en choisissant à chaque étape une arête uniformément parmi les arêtes de l’arbre pré-existant et greffant sur son « milieu » k-1 nouvelles arêtes. Lorsque k=2, cette procédure correspond à un algorithme introduit par Rémy. Pour chaque entier k2, nous décrivons la limite d’échelle de ces arbres lorsque le nombre d’étapes n tend vers l’infini : ils grandissent à la vitesse n 1/k vers un arbre réel aléatoire k-aire qui appartient à la famille des arbres de fragmentation auto-similaires. Cette convergence a lieu en probabilité, pour la topologie de Gromov–Hausdorff–Prokhorov. Nous étudions également l’emboîtement des arbres limites quand k varie.

DOI: 10.1214/14-AIHP622
Keywords: random growing trees, scaling limits, self-similar fragmentation trees, Gromov–Hausdorff–Prokhorov topology
     author = {Haas, B\'en\'edicte and Stephenson, Robin},
     title = {Scaling limits of $k$-ary growing trees},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
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Haas, Bénédicte; Stephenson, Robin. Scaling limits of $k$-ary growing trees. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 4, pp. 1314-1341. doi : 10.1214/14-AIHP622.

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