Persistent fluctuations of the swarm size of Brownian bees

TL;DR

This paper investigates large deviations and persistent fluctuations in the size of the swarm of Brownian bees, a model of branching Brownian particles, focusing on the probability of the swarm's maximum radius remaining within certain bounds over long times.

Contribution

It provides a detailed analysis of the large-deviation rate function for swarm size fluctuations, including an explicit solution for the one-dimensional case and asymptotic behaviors in all dimensions.

Findings

The probability density of swarm size deviations follows a large-deviation form with a rate function R_d(ell).

Explicit analytical form of the rate function is obtained for one-dimensional systems.

Asymptotic behaviors of the rate function are characterized for different regimes of ell relative to ell_0.

Abstract

The "Brownian bees" model describes a system of $N$ independent branching Brownian particles. At each branching event the particle farthest from the origin is removed, so that the number of particles remains constant at all times. Berestycki et al. (2020) proved that, at $N\to \infty$, the coarse-grained spatial density of this particle system lives in a spherically symmetric domain and is described by the solution of a free boundary problem for a deterministic reaction-diffusion equation. Further, they showed that, at long times, this solution approaches a unique spherically symmetric steady state with compact support: a sphere which radius $\ell_0$ depends on the spatial dimension $d$. Here we study fluctuations in this system in the limit of large $N$ due to the stochastic character of the branching Brownian motion, and we focus on persistent fluctuations of the swarm size. We evaluate the probability density $\mathcal{P}(\ell,N,T)$ that the maximum distance of a particle from the origin remains smaller than a specified value $\ell<\ell_0$, or larger than a specified value $\ell>\ell_0$, on a time interval $0<t<T$, where $T$ is very large. We argue that $\mathcal{P}(\ell,N,T)$ exhibits the large-deviation form $-\ln \mathcal{P} \simeq N T R_d(\ell)$. For all $d$ we obtain asymptotics of the rate function $R_d(\ell)$ in the regimes $\ell \ll \ell_0$, $\ell\gg \ell_0$ and $|\ell-\ell_0|\ll \ell_0$. For $d=1$ the whole rate function can be calculated analytically. We obtain these results by determining the optimal (most probable) density profile of the swarm, conditioned on the specified $\ell$, and by arguing that this density profile is spherically symmetric with its center at the origin.