The shape of multidimensional Brunet--Derrida particle systems

2013/05/01 by Berestycki, Nathanael, Zhao, Lee Zhuo · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1305.0254

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 → \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.

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