Probabilistic picture for particle number densities in stretched tips of the branching Brownian motion

TL;DR

This paper develops a stochastic model for the distribution of particle numbers near the leading particle in branching Brownian motion, deriving a formula for the probability density and estimating a key constant.

Contribution

It introduces a probabilistic framework to compute the particle number density near the tip of branching Brownian motion and determines an undetermined constant in the density decay.

Findings

Derived the probability density of particle numbers near the front

Confirmed the exponential decay factor in the particle density distribution

Provided an estimate for the previously unknown constant  in the decay

Abstract

In the framework of a stochastic picture for the one-dimensional branching Brownian motion, we compute the probability density of the number of particles near the rightmost one at a time $T$, that we take very large, when this extreme particle is conditioned to arrive at a predefined position $x_T$ chosen far ahead of its expected position $m_T$. We recover the previously-conjectured fact that the typical number density of particles a distance $\Delta$ to the left of the lead particle, when both $\Delta$ and $x_T-\Delta-m_T$ are large, is smaller than the mean number density by a factor proportional to $e^{-\zeta\Delta^{2/3}}$, where $\zeta$ is a constant that was so far undetermined. Our picture leads to an expression for the probability density of the particle number, from which a value for $\zeta$ may be inferred.