Local limits of galton-watson trees conditioned on the number of protected nodes

Romain Abraham (1) , Aymen Bouaziz (2) , Jean-François Delmas (3)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 Institut préparatoire aux études scientifiques et teckniques
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton-Watson tree conditioned on having a large number of marked vertices converges in distribution to the associated size-biased tree. We then apply this result to give the limit in distribution of a critical Galton-Watson tree conditioned on having a large number of protected nodes.

Domaines: Probabilités [math.PR]; Electrons fortement corrélés [cond-mat.str-el]; Physique Atomique [physics.atom-ph]

Romain Abraham, Aymen Bouaziz, Jean-François Delmas. Local limits of galton-watson trees conditioned on the number of protected nodes. 2015. ⟨hal-01195701v1⟩