## View abstract

### Session S04 - Random Walks and Related Topics

Thursday, July 15, 11:00 ~ 11:30 UTC-3

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

### Sarah Penington

#### University of Bath, United Kingdom   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakd748088b0a715442f81eec85a0c02305').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyd748088b0a715442f81eec85a0c02305 = 's.p&#101;n&#105;ngt&#111;n' + '&#64;'; addyd748088b0a715442f81eec85a0c02305 = addyd748088b0a715442f81eec85a0c02305 + 'b&#97;th' + '&#46;' + '&#97;c' + '&#46;' + '&#117;k'; var addy_textd748088b0a715442f81eec85a0c02305 = 's.p&#101;n&#105;ngt&#111;n' + '&#64;' + 'b&#97;th' + '&#46;' + '&#97;c' + '&#46;' + '&#117;k';document.getElementById('cloakd748088b0a715442f81eec85a0c02305').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyd748088b0a715442f81eec85a0c02305 + '\'>'+addy_textd748088b0a715442f81eec85a0c02305+'<\/a>';

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 about 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, building on earlier work of J. Bérard and P. Maillard. 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.

Joint work with Matt Roberts (University of Bath) and Zsófia Talyigás (University of Bath).