$N$-Branching random walk with $\alpha$-stable spine

We consider a branching-selection particle system on the real line, introduced by Brunet and Derrida in [Phys. Rev. E, 56 (1997), pp. 2597–2604]. In this model the size of the population is fixed to a constant $N$. At each step individuals in the population reproduce independently, making children around their current position. Only the $N$ rightmost children survive to reproduce at the next step. Bérard and Gouéré studied the speed at which the cloud of individuals drifts in [Comm. Math. Phys., 298 (2010), pp. 323–342], assuming the tails of the displacement decays at exponential rate; Bérard and Maillard [Electron. J. Probab., 19 (2014), 22] took interest in the case of heavy tail displacements. We take interest in an intermediate model, considering branching random walks in which the critical “spine” behaves as an $\alpha$-stable random walk.