Limit theorems for conditioned non-generic Galton–Watson trees

Kortchemski, Igor

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 489-511.

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.

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.

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