Quenched Survival of Bernoulli Percolation on Galton–Watson Trees

We explore the survival function for 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 about the behavior of the random function g(𝐓, · ) , where 𝐓 is drawn from the Galton–Watson distribution. These include almost sure smoothness in the supercritical region; an expression for the kth -order Taylor expansion of g(𝐓, · ) at criticality in terms of limits of martingales defined from 𝐓 (this requires a moment condition depending on k ); and a proof that the kth order derivative extends continuously to the critical value. Each of these results is shown to hold for almost every Galton–Watson tree.