  We consider a system of $N$ particles on the real line that evolves through
iteration of the following steps: 1) every particle splits into two, 2) each
particle jumps according to a prescribed displacement distribution supported on
the positive reals and 3) only the $N$ right-most particles are retained, the
others being removed from the system. This system has been introduced in the
physics literature as an example of a microscopic stochastic model describing
the propagation of a front. Its behavior for large $N$ is now well understood
-- both from a physical and mathematical viewpoint -- in the case where the
displacement distribution admits exponential moments. Here, we consider the
case of displacements with regularly varying tails, where the relevant space
and time scales are markedly different. We characterize the behavior of the
system for two distinct asymptotic regimes. First, we prove convergence in law
of the rescaled positions of the particles on a time scale of order $\log N$
and give a construction of the limit based on the records of a space-time
Poisson point process. Second, we determine the appropriate scaling when we let
first the time horizon, then $N$ go to infinity.
