Genealogy and spatial distribution of the $N$-particle branching random walk with polynomial tails

TL;DR

This paper investigates the long-term behavior of a large $N$-particle branching random walk with selection, revealing genealogical structures and spatial concentration when jumps have polynomial tails.

Contribution

It provides a detailed analysis of the genealogy and spatial distribution of the system with polynomial tail jumps, extending understanding of such processes.

Findings

Genealogy follows a star-shaped coalescent at large times.

Most particles cluster near the leftmost particle.

Results apply to systems with regularly varying tail jump distributions.

Abstract

The $N$-particle branching random walk is a discrete time branching particle system with selection. We have $N$ particles located on the real line at all times. At every time step each particle is replaced by two offspring, and each offspring particle makes a jump of non-negative size 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. Inspired by work of J. B\'erard and P. Maillard, we examine the long term behaviour of this particle system in the case where the jump distribution has regularly varying tails and the number of particles is large. We prove that at a typical large time the genealogy of the population is given by a star-shaped coalescent, and that almost the whole population is near the leftmost particle on the relevant space scale.