The distance between the two BBM leaders

TL;DR

This paper investigates the asymptotic behavior of the distance between the two rightmost particles in branching Brownian motion, revealing an algebraic correction to previously known exponential tail estimates.

Contribution

The authors derive sharp asymptotics for the tail probability of the distance between the two leading particles, including an algebraic correction term.

Findings

Established precise tail asymptotics for the distance

Discovered an algebraic correction to exponential tail behavior

Connected the problem to PDEs related to Fisher--KPP equation

Abstract

We study the distance between the two rightmost particles in branching Brownian motion. Derrida and the second author have shown that the long-time limit $d_{12}$ of this random variable can be expressed in terms of PDEs related to the Fisher--KPP equation. We use such a representation to determine the sharp asymptotics of $\mathbb{P}(d_{12} > a)$ as $a\to+\infty$. These tail asymptotics were previously known to "exponential order;" we discover an algebraic correction to this behavior.