Tail asymptotics for the total progeny of the critical killed branching random walk

TL;DR

This paper analyzes the tail behavior of the total progeny in a critical killed branching random walk, confirming a predicted asymptotic decay rate of (n ln^2(n))^{-1}.

Contribution

It provides a rigorous proof of the tail asymptotics for the total progeny in a critical killed branching random walk, validating previous predictions.

Findings

Tail distribution of total progeny decays as (n ln^2(n))^{-1}

Confirms the asymptotic behavior predicted by Addario-Berry and Broutin

Provides a mathematical foundation for understanding critical killed branching processes

Abstract

We consider a branching random walk on $\mathbb{R}$ with a killing barrier at zero. At criticality, the process becomes eventually extinct, and the total progeny $Z$ is therefore finite. We show that the tail distribution of $Z$ displays a typical behaviour in $(n\ln^2(n))^{-1}$, which confirms the prediction of Addario-Berry and Broutin.