Probabilities of moderately atypical fluctuations of the size of a swarm of Brownian bees

The Brownian bees model describes an ensemble of N = const independent branching Brownian particles. The conservation of N is provided by a modified branching process. When a particle branches into two particles, the particle which is farthest from the origin is eliminated simultaneously. The spatial density of the particles is governed by the solution of a free boundary problem for a reaction-diffusion equation in the limit of N >> 1. At long times, the particle density approaches a spherically symmetric steady-state solution with a compact support of radius (l(0)) over bar. However, at finite N, the radius of this support, L, fluctuates. The variance of these fluctuations appears to exhibit a logarithmic anomaly [Siboni et al., Phys. Rev. E 104, 054131 (2021)]. It is proportional to N-1 lnN at N -> infinity. We investigate here the tails of the probability density function (PDF), P(L), of the swarm radius, when the absolute value of the radius fluctuation Delta L = L - (l(0)) over bar is sufficiently larger than the typical fluctuations' scale determined by the variance. For negative deviations the PDF can be obtained in the framework of the optimal fluctuation method. This part of the PDF displays the scaling behavior ln P proportional to -N Lambda L-2 ln(-1)(Lambda L-2), demonstrating a logarithmic anomaly at small negative Delta L. For the opposite sign of the fluctuation, Delta L > 0, the PDF can be obtained with an approximation of a single particle, running away. We find that ln P proportional to -N-1/2 Delta L. We consider in this paper only the case when vertical bar Delta L vertical bar is much less than the typical radius of the swarm at N >> 1.

Automated summary

The Brownian bees model describes a collection of N independent branching Brownian particles. The density of the particles is governed by a reaction-diffusion equation. The particle density approaches a symmetric steady-state solution at long times, but fluctuates at finite N values.