Local limits of conditioned Galton-Watson trees II: the condensation case

TL;DR

This paper studies the local limits of conditioned Galton-Watson trees, focusing on the condensation case where the limit features a node with infinite out-degree, complementing previous work on the generic case.

Contribution

It provides a complete analysis of the non-generic condensation case for Galton-Watson trees conditioned on large out-degree sets.

Findings

Characterization of the condensation limit tree with an infinite out-degree node

Distinction between generic and non-generic local limits

Extension of previous results to the condensation scenario

Abstract

We provide a complete picture of the local convergence of critical or subcritical Galton-Watson tree conditioned on having a large number of individuals with out-degree in a given set. The generic case, where the limit is a random tree with an infinite spine has been treated in a previous paper. We focus here on the non-generic case, where the limit is a random tree with a node with infinite out-degree. This case corresponds to the so-called condensation phenomenon.