Very fat geometric Galton-Watson trees

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 nth generation and (an, 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 ∗) 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 ∗) increases rapidly, a condensation phenomenon appears and the root of the limiting tree has an infinite number of offspring.