Branching random walk with selection at critical rate

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 $\exp\left(a n^{1/3}\right)$\mathrm{.}Ateachstep$n$, every individual dies while reproducing independently, making children around their current position according to i.i.d. point processes. Only the $\exp (a(n+1{)}^{1\mathrm{/}3})$ rightmost children survive to form the $(n+1{)}^{\mathrm{t}\mathrm{h}}$ 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.