Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Local limits of large Galton–Watson trees rerooted at a random vertex

Benedikt Stufler

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We discuss various forms of convergence of the vicinity of a uniformly at random selected vertex in random simply generated trees, as the size tends to infinity. For the standard case of a critical Galton–Watson tree conditioned to be large the limit is the invariant random sin-tree constructed by Aldous (1991). In the condensation regime, we describe in complete generality the asymptotic local behaviour from a random vertex up to its first ancestor with large degree. Beyond this distinguished ancestor, different behaviour may occur, depending on the branching weights. In a subregime of complete condensation, we obtain convergence toward a novel limit tree, that describes the asymptotic shape of the vicinity of the full path from a random vertex to the root vertex. This includes the case where the offspring distribution follows a power law up to a factor that varies slowly at infinity.


Nous discutons de plusieurs formes de convergence du voisinage d’un sommet aléatoire uniforme dans des arbres aléatoires simplement générés, lorsque leur taille tend vers l’infini. Pour le cas standard d’un arbre de Galton–Watson critique conditionné à être grand, la limite est le sin-tree invariant aléatoire construit par Aldous (1991). Dans le régime de condensation, nous décrivons en toute généralité le comportement asymptotique local depuis un sommet aléatoire jusqu’à son premier ancêtre de grand degré. Au delà de cet ancêtre distingué, différents comportements peuvent apparaître selon les poids de branchement. Dans un sous-régime de condensation complète, nous obtenons la convergence vers un nouvel arbre limite, qui décrit la forme asymptotique du voisinage du chemin complet depuis un sommet aléatoire jusqu’à la racine. Cela inclut le cas où la distribution de la descendance suit une loi de puissance, à un facteur près qui varie lentement à l’infini.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 1 (2019), 155-183.

Received: 5 November 2016
Revised: 19 December 2017
Accepted: 21 December 2017
First available in Project Euclid: 18 January 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60B10: Convergence of probability measures

Local weak limits Simply generated trees Fringe distributions


Stufler, Benedikt. Local limits of large Galton–Watson trees rerooted at a random vertex. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 1, 155--183. doi:10.1214/17-AIHP879.

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