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We consider a system of $N$ particles on the real line that evolves throughiteration of the following steps: 1) every particle splits into two, 2) eachparticle jumps according to a prescribed displacement distribution supported onthe positive reals and 3) only the $N$ right-most particles are retained, theothers being removed from the system. This system has been introduced in thephysics literature as an example of a microscopic stochastic model describingthe propagation of a front. Its behavior for large $N$ is now well understood-- both from a physical and mathematical viewpoint -- in the case where thedisplacement distribution admits exponential moments. Here, we consider thecase of displacements with regularly varying tails, where the relevant spaceand time scales are markedly different. We characterize the behavior of thesystem for two distinct asymptotic regimes. First, we prove convergence in lawof 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-timePoisson point process. Second, we determine the appropriate scaling when we letfirst the time horizon, then $N$ go to infinity.

The limiting process of $N$-particle branching random walk with polynomial tails