Haas, Bénédicte; Stephenson, Robin
Scaling limits of k-ary growing trees
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4 , p. 1314-1341
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MR 3414449
doi : 10.1214/14-AIHP622
URL stable : http://www.numdam.org/item?id=AIHPB_2015__51_4_1314_0

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.
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.

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