Very fat geometric galton-watson trees

TL;DR

This paper investigates the local limits of geometric Galton-Watson trees conditioned on their n-th generation size, revealing three regimes: infinite spine, infinite skeleton, and condensation, depending on the growth rate of the conditioning sequence.

Contribution

It characterizes the asymptotic local structures of conditioned geometric Galton-Watson trees across different growth regimes of the conditioning sequence.

Findings

Identifies three distinct local limit regimes based on growth rate.

Describes the structure of the limit trees in each regime.

Shows the emergence of a condensation phenomenon with rapid growth.

Abstract

Let $\tau$n 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 $\in$ N *) is a deterministic positive sequence. We study the local limit of these trees $\tau$n as n $\rightarrow$ $\infty$ and observe three distinct regimes: if (an, n $\in$ 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 $\in$ N *) increases rapidly, a condensation phenomenon appears and the root of the limiting tree has an infinite number of offspring.