Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion

TL;DR

This paper analyzes the probability distribution of the empirical velocity of the rightmost particle in a one-dimensional branching Brownian motion, providing detailed asymptotics and generalizations for long time behavior.

Contribution

It determines the prefactor in the probability distribution of the rightmost particle's velocity, extending previous exponential estimates to precise asymptotics and generalizations.

Findings

Derived the asymptotic form of the probability distribution for the rightmost particle's velocity.

Identified the prefactor in the distribution beyond the exponential decay.

Extended results to multiple seed particles and related branching random walks.

Abstract

Consider a one-dimensional branching Brownian motion, and rescale the coordinate and time so that the rates of branching and diffusion are both equal to $1$. If $X_1(t)$ is the position of the rightmost particle of the branching Brownian motion at time $t$, the empirical velocity $c$ of this rightmost particle is defined as $c=X_1(t)/t$. Using the Fisher-KPP equation, we evaluate the probability distribution ${\mathcal P(c,t)}$ of this empirical velocity $c$ in the long time $t$ limit for $c > 2$. It was already known that, for a single seed particle, ${\mathcal P(c,t)} \sim \exp \,[-(c^2/4-1)t]$ up to a prefactor that can depend on $c$ and $t$. Here we show how to determine this prefactor. The result can be easily generalized to the case of multiple seed particles and to branching random walks associated to other traveling wave equations.