Fluctuations of a swarm of Brownian bees

The “Brownian bees” model is a new member of a family of Brunet-Derrida particle systems which mimic different aspects of biological selection. The model describes an ensemble of $N$ independent branching Brownian particles. When a particle branches into two particles, the particle farthest from the origin is eliminated so as to keep the number of particles constant. In the limit of $N \to \infty$, the coarse-grained particle density is governed by the solution of a free boundary problem for a simple reaction-diffusion equation. At long times the particle density approaches a spherically symmetric steady-state solution with a compact support. We studied fluctuations of the “swarm of bees” due to the random character of the branching Brownian motion in the limit of large but finite $N$. We considered a one-dimensional setting and focused on two fluctuating quantities: the swarm center of mass $X(t)$ and the swarm radius $l(t)$. Unsurprisingly, the variance of $X (t)$ scales as $\frac{1}{N}$. The variance of $l(t)$, however, exhibits an anomalous scaling $\frac{\ln N}{N}$. This anomaly appears because all spatial scales, including a narrow region near the edges of the swarm (where only a few particles are present), contribute to the variance. I will also briefly discuss some large-deviation properties of the model.

Work done in collaboration with Maor Siboni and Pavel Sasorov

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