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Jean-François Le Gall
Random real trees
Annales de la faculté des sciences de Toulouse Sér. 6, 15 no. 1 (2006), p. 35-62, doi: 10.5802/afst.1112
Article PDF | Analyses MR 2225746 | Zbl 1129.60047

Résumé - Abstract

Nous discutons certains développements récents de la théorie des arbres réels aléatoires, dont le prototype est le CRT introduit par Aldous en 1991. Nous introduisons le formalisme d’arbre réel, qui fournit une présentation élégante de la théorie, et en particulier des relations entre les arbres de Galton-Watson discrets et les arbres continus aléatoires. Nous discutons ensuite la classe des arbres auto-similaires appelés arbres stables, qui généralisent le CRT. Nous présentons plusieurs résultats importants au sujet des arbres stables, notamment leur propriété de branchement, analogue continu d’une propriété bien connue pour les arbres de Galton-Watson, et le calcul de leurs dimensions fractales. Nous considérons ensuite les arbres spatiaux, qui combinent la structure généalogique d’un arbre réel avec des déplacements dans l’espace, et nous expliquons leurs liens avec les superprocessus. Dans la dernière partie, nous traitons un conditionnement particulier des arbres spatiaux, qui est étroitement lié à certains résultats asymptotiques pour les quadrangulations planes aléatoires.

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