Probabilities of moderately atypical fluctuations of the size of a swarm of Brownian Bees
Pavel Sasorov, Arkady Vilenkin, Naftali R. Smith

TL;DR
This paper analyzes the probability of atypical fluctuations in the size of a swarm of Brownian bees, revealing a logarithmic anomaly in the tail distribution for negative deviations and a different scaling for positive deviations.
Contribution
It provides a detailed analysis of the tail behavior of the swarm radius distribution, uncovering a logarithmic anomaly for negative fluctuations and a new scaling law for positive fluctuations.
Findings
Negative deviations follow a scaling: ln P ∝ -N ΔL^2 / ln(ΔL^{-2})
Positive deviations follow a scaling: ln P ∝ -N^{1/2} ΔL
The study characterizes the probability density function tails for finite N.
Abstract
The ``Brownian bees'' model describes an ensemble of ~const independent branching Brownian particles. The conservation of 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 . At long times, the particle density approaches a spherically symmetric steady state solution with a compact support of radius . However, at finite , the radius of this support, , fluctuates. The variance of these fluctuations appears to exhibit a logarithmic anomaly [Siboni {\em et al}., Phys. Rev. E. {\bf104}, 054131 (2021)]. It is proportional to at . We investigate here the tails of the…
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Taxonomy
TopicsStochastic processes and statistical mechanics · Theoretical and Computational Physics · Complex Network Analysis Techniques
