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

Abstract

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.