Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Limit theorems for conditioned non-generic Galton–Watson trees

Igor Kortchemski

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Abstract

We study a particular type of subcritical Galton–Watson trees, which are called non-generic trees in the physics community. In contrast with the critical or supercritical case, it is known that condensation appears in certain large conditioned non-generic trees, meaning that with high probability there exists a unique vertex with macroscopic degree comparable to the total size of the tree. Using recent results concerning subexponential distributions, we investigate this phenomenon by studying scaling limits of such trees and show that the situation is completely different from the critical case. In particular, the height of such trees grows logarithmically in their size. We also study fluctuations around the condensation vertex.

Résumé

Nous étudions une classe particulière d’arbres de Galton–Watson sous-critiques, appelés arbres non-génériques en physique. Contrairement au cas critique ou surcritique, il est connu qu’une condensation apparaît dans certains grands arbres non-génériques conditionnés, c’est-à-dire qu’avec grande probabilité il existe un unique sommet de degré macroscopique comparable à la taille totale de l’arbre. En utilisant des résultats récents relatifs à des lois sousexponentielles, nous étudions ce phénomène en étudiant les limites d’échelles de tels arbres et montrons que la situation est complètement différente du cas critique. En particulier, la hauteur de ces arbres croît logarithmiquement en leur taille. Nous étudions aussi les fluctuations autour du sommet de condensation.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 51, Number 2 (2015), 489-511.

Dates
First available in Project Euclid: 10 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1428672678

Digital Object Identifier
doi:10.1214/13-AIHP580

Mathematical Reviews number (MathSciNet)
MR3335012

Zentralblatt MATH identifier
1315.60091

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F17: Functional limit theorems; invariance principles
Secondary: 05C80: Random graphs [See also 60B20] 05C05: Trees

Keywords
Condensation Subcritical Galton–Watson trees Scaling limits Subexponential distributions

Citation

Kortchemski, Igor. Limit theorems for conditioned non-generic Galton–Watson trees. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 2, 489--511. doi:10.1214/13-AIHP580. https://projecteuclid.org/euclid.aihp/1428672678


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