Very fat geometric Galton-Watson trees

¹ Institut Denis Poisson, Université d’Orléans, Université de Tours, CNRS, B.P. 6759, 45067 Orléans cedex 2, France.
² Institut Préparatoire aux Études Scientifiques et Techniques, La Marsa, 2070 - Tunis.
³ Université Paris-Est, École des Ponts, CERMICS, 6-8 av. Blaise Pascal, Champs-sur-Marne, 77455 Marne La Vallée, France.

^* Corresponding author: delmas@cermics.enpc.fr

Received: 5 October 2018
Accepted: 12 November 2019

Abstract

Let τ_n be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on {Z_n = a_n} where Z_n is the size of the nth generation and (a_n, n ∈ ℕ^*) is a deterministic positive sequence. We study the local limit of these trees τ_n as n →∞ and observe three distinct regimes: if (a_n, 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 (a_n, n ∈ ℕ^*) increases rapidly, a condensation phenomenon appears and the root of the limiting tree has an infinite number of offspring.