Quenched Survival of Bernoulli Percolation on Galton-Watson Trees

Marcus Michelen, Robin Pemantle, Josh Rosenberg

Published 2018-05-09Version 1

We explore the subject of percolation on Galton-Watson trees. Letting $g(T,p)$ represent the probability a tree $T$ survives Bernoulli percolation with parameter $p$, we establish several results relating to the behavior of $g$ in the supercritical region. These include an expression for the third order Taylor expansion of $g$ at criticality in terms of limits of martingales depending on $T$, a proof that $g$ is smooth on the supercritical region, and a proof that $g'$ extends continuously to the boundary of the supercritical region. Allowing for some mild moment constraints on the offspring distribution, each of these results is shown to hold for almost every Galton-Watson tree.