Barycentric Brownian Bees

TL;DR

This paper proves an invariance principle for the barycenter of a particle system undergoing branching Brownian motion with a selection mechanism in multiple dimensions, demonstrating Harris recurrence of the process.

Contribution

It introduces a new invariance principle for the barycenter in a Brunet-Derrida particle system with selection in higher dimensions.

Findings

Established Harris recurrence for the process from its barycenter.

Proved an invariance principle for the barycenter in a multi-dimensional setting.

Analyzed the dynamics of the particle system with selection mechanism.

Abstract

We establish an invariance principle for the barycenter of a Brunet-Derrida particle system in $d$ dimensions. The model consists of $N$ particles undergoing dyadic branching Brownian motion with rate $1$. At a branching event, the number of particles is kept equal to $N$ by removing the particle located furthest away from the barycenter. To prove the invariance principle, a key step is to establish Harris recurrence for the process viewed from its barycenter.