Sarah Penington (University of Bath, United Kingdom)
Title: Genealogy of the N-particle branching random walk with polynomial tails
Abstract: The N-particle branching random walk is a discrete time branching particle system with selection consisting of N particles located on the real line. At every time step, each particle is replaced by two offspring, and each offspring particle makes a jump from its parent’s location, independently from the other jumps, according to a given jump distribution. Then only the N rightmost particles survive; the other particles are removed from the system to keep the population size constant.
I will discuss recent results and open conjectures about the long-term behaviour of this particle system when N, the number of particles, is large. In the case where the jump distribution has regularly varying tails, building on earlier work of J. Bérard and P. Maillard, we prove that at a typical large time the genealogy is given by a star-shaped coalescent, and that almost the whole population is near the leftmost particle on the relevant space scale.