THE SHAPE OF MULTIDIMENSIONAL BRUNET-DERRIDA PARTICLE SYSTEMS

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 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.