Maximal displacement in the $d$-dimensional branching Brownian motion

TL;DR

This paper analyzes the maximal displacement in d-dimensional branching Brownian motion, revealing a dimension-dependent logarithmic correction and a geometric property of individuals on the frontier.

Contribution

It provides a new asymptotic description of maximal displacement in multi-dimensional branching Brownian motion, highlighting the role of dimension and geometric relationships.

Findings

Maximal displacement follows a ballistic order plus a dimension-dependent logarithmic correction.

Individuals on the frontier are close parents if and only if they are geographically close, with high probability.

The proof uses simple geometrical arguments.

Abstract

We consider a branching Brownian motion evolving in $\mathbb{R}^d$. We prove that the asymptotic behaviour of the maximal displacement is given by a first ballistic order, plus a logarithmic correction that increases with the dimension $d$. The proof is based on simple geometrical evidence. It leads to the interesting following side result: with high probability, for any $d \geq 2$, individuals on the frontier of the process are close parents if and only if they are geographically close.