  We consider a branching-selection particle system on the real line,
introduced by Brunet and Derrida. 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. B\'erard and
Gou\'er\'e studied the speed at which the cloud of individuals drifts, assuming
the tails of the displacement decays at exponential rate; B\'erard and Maillard
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.
