## The Annals of Applied Probability

### The shape of multidimensional Brunet–Derrida particle systems

#### Abstract

We introduce particle systems in one or more dimensions in which particles perform branching Brownian motion and the population size is kept constant equal to $N>1$, through the following selection mechanism: at all times only the $N$ fittest particles survive, while all the other particles are removed. Fitness is measured with respect to some given score function $s:\mathbb{R}^{d}\to\mathbb{R}$. For some choices of the function $s$, it is proved that the cloud of particles travels at positive speed in some possibly random direction. In the case where $s$ is linear, we show under some mild assumptions that the shape of the cloud scales like $\log N$ in the direction parallel to motion but at least $(\log N)^{3/2}$ in the orthogonal direction. We conjecture that the exponent $3/2$ is sharp. In order to prove this, we obtain the following result of independent interest: in one-dimensional systems, the genealogical time is greater than $c(\log N)^{3}$. We discuss several open problems and explain how our results can be viewed as a rigorous justification in our setting of empirical observations made by Burt [Evolution 54 (2000) 337–351] in support of Weismann’s arguments for the role of recombination in population genetics.

#### Article information

Source
Ann. Appl. Probab., Volume 28, Number 2 (2018), 651-687.

Dates
Revised: August 2014
First available in Project Euclid: 11 April 2018

https://projecteuclid.org/euclid.aoap/1523433619

Digital Object Identifier
doi:10.1214/14-AAP1062

Mathematical Reviews number (MathSciNet)
MR3784486

Zentralblatt MATH identifier
06897941

#### Citation

Berestycki, Nathanaël; Zhao, Lee Zhuo. The shape of multidimensional Brunet–Derrida particle systems. Ann. Appl. Probab. 28 (2018), no. 2, 651--687. doi:10.1214/14-AAP1062. https://projecteuclid.org/euclid.aoap/1523433619

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