The shape of multidimensional Brunet--Derrida particle systems

TL;DR

This paper studies multidimensional particle systems with branching Brownian motion and selection, revealing how the shape of the particle cloud scales with population size and providing insights into genealogical times and population genetics.

Contribution

It introduces a multidimensional model with selection, proves shape scaling laws, and links these results to genealogical times and genetic recombination theories.

Findings

Particle clouds travel at positive speed in certain directions.

Shape scales like log N parallel to motion and at least (log N)^{3/2} orthogonally.

Genealogical time exceeds c(log N)^3 in one-dimensional 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 $s:\R^d \to \R$. 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 assumptions on the initial configuration that the shape of the cloud scales like $\log N$ in the direction parallel to motion but at least $c(\log N)^{3/2}$ in the orthogonal direction for some $c > 0$. We conjecture that the exponent 3/2 is sharp. This result is equivalent to the following result of independent interest: in one-dimensional systems, the genealogical time is greater than $c(\log N)^3$, thereby contributing a step towards the original predictions of Brunet and Derrida. We discuss several open problems and also explain how our results can be viewed as a rigorous justification of Weismann's arguments for the role of recombination in population genetics.