## 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.Bérard 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.

Original language | English |
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Article number | 93 |

Pages (from-to) | 1-65 |

Journal | Electronic Journal of Probability |

Volume | 27 |

Early online date | 26 Jul 2022 |

DOIs | |

Publication status | Published - 31 Dec 2022 |

### Bibliographical note

Funding Information:*SP and MR are supported by Royal Society University Research Fellowships. ZT is supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under the project EP/L015684/1. †University of Bath, UK. E-mail: [email protected] ‡University of Bath, UK. E-mail: [email protected] §University of Bath, UK. E-mail: [email protected]

Publisher Copyright:

© 2022, Institute of Mathematical Statistics. All rights reserved.

## Keywords

- branching random walk
- genealogy
- heavy-tailed distribution
- selection
- star-shaped

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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