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.