Percolation On Galton-Watson Trees

We consider both Bernoulli and invasion percolation on Galton-Watson trees. In the former case, we show that the quenched survival function is smooth on the supercritical window and smooth from the right at criticality. We also study critical percolation conditioned to reach depth $n$, and construct the incipient infinite cluster by taking $n \to \infty$; quenched limit theorems are proven for the asymptotic size of the layers of the incipient infinite cluster. In the case of invasion percolation, we show that the law of the unique ray in the invasion cluster is absolutely continuous with respect to the limit uniform measure. All results are under assumptions for the offspring distribution of the underlying Galton-Watson tree.