Conditioned Marked Galton-Watson trees

We consider a Galton–Watson tree in which each node is independently marked, with a probability that depends on its number of offspring.

We give a complete picture of the local convergence of critical or subcritical marked Galton–Watson trees, conditioned on having a large number of marks.

In certain cases, the limit is a randomly marked tree with an infinite spine, known as the marked Kesten tree. In other cases, the local limit is a randomly marked tree with a node having infinitely many children. This corresponds to the so-called marked condensation phenomenon.