Branching random walk with selection at critical rate

TL;DR

This paper studies a branching-selection particle system with a population size limited by an exponential function of time, analyzing the extremal particle positions through coupling with a killed branching random walk.

Contribution

It introduces a generalized model of branching random walk with selection at a critical rate and derives the asymptotic behavior of extremal particles.

Findings

Asymptotic position of extremal particles determined

Coupling with killed branching random walk provides key insights

Generalizes previous models of selection in branching processes

Abstract

We consider a branching-selection particle system on the real line. In this model the total size of the population at time $n$ is limited by $\exp\left(a n^{1/3}\right)$. At each step $n$, every individual dies while reproducing independently, making children around their current position according to i.i.d. point processes. Only the $\exp\left(a(n+1)^{1/3}\right)$ rightmost children survive to form the $(n+1)^\mathrm{th}$ generation. This process can be seen as a generalisation of the branching random walk with selection of the $N$ rightmost individuals, introduced by Brunet and Derrida. We obtain the asymptotic behaviour of position of the extremal particles alive at time $n$ by coupling this process with a branching random walk with a killing boundary.