Branching random walk with selection at critical rate

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.

Updated version

References in corpus (2)

• Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization
• $N$-Branching random walk with $\alpha$-stable spine

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