## Bernoulli

• Bernoulli
• Volume 23, Number 3 (2017), 1784-1821.

### Branching random walk with selection at critical rate

Bastien Mallein

#### 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 $\exp (an^{1/3})$. 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 $\exp (a(n+1)^{1/3})$ rightmost children survive to form the $(n+1)$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 (Phys. Rev. E (3) 56 (1997) 2597–2604). 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.

#### Article information

Source
Bernoulli, Volume 23, Number 3 (2017), 1784-1821.

Dates
Revised: November 2015
First available in Project Euclid: 17 March 2017

https://projecteuclid.org/euclid.bj/1489737625

Digital Object Identifier
doi:10.3150/15-BEJ796

Mathematical Reviews number (MathSciNet)
MR3624878

Zentralblatt MATH identifier
06714319

Keywords
branching random walk selection

#### Citation

Mallein, Bastien. Branching random walk with selection at critical rate. Bernoulli 23 (2017), no. 3, 1784--1821. doi:10.3150/15-BEJ796. https://projecteuclid.org/euclid.bj/1489737625

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