Quenched Survival of Bernoulli Percolation on Galton-Watson Trees

TL;DR

This paper investigates the behavior of the survival probability function for Bernoulli percolation on Galton-Watson trees, revealing smoothness properties and detailed Taylor expansions at criticality, applicable to almost all such trees.

Contribution

It provides new results on the smoothness and Taylor expansion of the survival function at criticality for percolation on Galton-Watson trees, including martingale-based expressions.

Findings

Almost sure smoothness in the supercritical region

Explicit Taylor expansion at criticality

Continuity of derivatives at critical point

Abstract

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 $g(\mathbf{T} , \cdot)$, where $\mathbf{T}$ is drawn from the Galton-Watson distribution. These include almost sure smoothness in the supercritical region; an expression for the $k\text{th}$-order Taylor expansion of $g(\mathbf{T} , \cdot)$ at criticality in terms of limits of martingales defined from $\mathbf{T}$ (this requires a moment condition depending on $k$); and a proof that the $k\text{th}$ order derivative extends continuously to the critical value. Each of these results is shown to hold for almost every Galton-Watson tree.