Local limits of conditioned Galton-Watson trees: The condensation case

We provide a complete picture of the local convergence of critical or sub-critical Galton-Watson trees 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 local limit is a random tree with a node with infinite out-degree. This case corresponds to the so-called condensation phenomenon.