Very fat geometric galton-watson trees

Abstract : Let τn be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on {Zn = an} where Zn is the size of the n-th generation and (an, n ∈ N *) is a deterministic positive sequence. We study the local limit of these trees τn as n → ∞ and observe three distinct regimes: if (an, n ∈ N *) grows slowly, the limit consists in an infinite spine decorated with finite trees (which corresponds to the size-biased tree for critical or subcritical offspring distributions), in an intermediate regime, the limiting tree is composed of an infinite skeleton (that does not satisfy the branching property) still decorated with finite trees and, if the sequence (an, n ∈ N *) increases rapidly, a condensation phenomenon appears and the root of the limiting tree has an infinite number of offspring.
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Pré-publication, Document de travail
2017
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https://hal.archives-ouvertes.fr/hal-01591656
Contributeur : Romain Abraham <>
Soumis le : jeudi 21 septembre 2017 - 17:13:40
Dernière modification le : mardi 10 octobre 2017 - 13:47:45

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  • HAL Id : hal-01591656, version 1
  • ARXIV : 1709.09403

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Romain Abraham, Aymen Bouaziz, Jean-François Delmas. Very fat geometric galton-watson trees. 2017. 〈hal-01591656〉

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