Open Access
August 2017 Branching random walk with selection at critical rate
Bastien Mallein
Bernoulli 23(3): 1784-1821 (August 2017). DOI: 10.3150/15-BEJ796


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.


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Bastien Mallein. "Branching random walk with selection at critical rate." Bernoulli 23 (3) 1784 - 1821, August 2017.


Received: 1 March 2015; Revised: 1 November 2015; Published: August 2017
First available in Project Euclid: 17 March 2017

zbMATH: 06714319
MathSciNet: MR3624878
Digital Object Identifier: 10.3150/15-BEJ796

Keywords: Branching random walk , selection

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

Vol.23 • No. 3 • August 2017
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