Survival of inhomogeneous Galton-Watson processes

TL;DR

This paper investigates the survival characteristics of inhomogeneous Galton-Watson processes, determining the branching number and analyzing how survival probabilities and percolation clusters behave across generations.

Contribution

It introduces the concept of the branching number for inhomogeneous Galton-Watson trees and explores the uniform behavior of survival probabilities under percolation.

Findings

The branching number is almost surely constant.

Survival probability varies smoothly between generations.

Percolation clusters grow at rates uniform in retention probability p.

Abstract

We study survival properties of inhomogeneous Galton-Watson processes. We determine the so-called branching number (which is the reciprocal of the critical value for percolation) for these random trees (conditioned on being infinite), which turns out to be an a.s.\ constant. We also shed some light on the way the survival probability varies between the generations. When we perform independent percolation on the family tree of an inhomogeneous Galton-Watson process, the result is essentially a family of inhomogeneous Galton-Watson processes, parametrized by the retention probability $p$. We provide growth rates, uniformly in $p$, of the percolation clusters, and also show uniform convergence of the survival probability from the $n$-th level along subsequences. These results also establish, as a corollary, the supercritical continuity of the percolation function. Some of our results are generalisations of results by Lyons (1992).