Persistent fluctuations of the swarm size of Brownian bees

The ``Brownian bees'' model describes a system of $N$-independent branching Brownian particles. At each event the particle farthest from origin is removed so that number particles remains constant at all times. Berestycki et al. [arXiv:2006.06486] proved $N\ensuremath{\rightarrow}\ensuremath{\infty}$ coarse-grained spatial density this lives in spherically symmetric domain and described by solution free boundary problem for deterministic reaction-diffusion equation. Furthermore, they showed [arXiv:2005.09384] that, long times, approaches unique steady state with compact support: sphere whose radius ${\ensuremath{\ell}}_{0}$ depends on dimension $d$. Here we study fluctuations limit large $N$ due to stochastic character motion, focus persistent swarm size. We evaluate probability $\mathcal{P}(\ensuremath{\ell},N,T)$ maximum distance smaller than specified value $\ensuremath{\ell}<{\ensuremath{\ell}}_{0}$ or larger $\ensuremath{\ell}>{\ensuremath{\ell}}_{0}$ time interval $0<t<T$, where $T$ very large. argue exhibits large-deviation form $\ensuremath{-}ln\phantom{\rule{0.16em}{0ex}}\mathcal{P}\ensuremath{\simeq}NT{R}_{d}(\ensuremath{\ell})$. For $d$'s obtain asymptotics rate function ${R}_{d}(\ensuremath{\ell})$ regimes $\ensuremath{\ell}\ensuremath{\ll}{\ensuremath{\ell}}_{0},\phantom{\rule{0.28em}{0ex}}\ensuremath{\ell}\ensuremath{\gg}{\ensuremath{\ell}}_{0}$, $|\ensuremath{\ell}\ensuremath{-}{\ensuremath{\ell}}_{0}|\ensuremath{\ll}{\ensuremath{\ell}}_{0}$. $d=1$ whole can be calculated analytically. these results determining optimal (most probable) profile swarm, conditioned $\ensuremath{\ell}$ arguing its center origin.