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(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.