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

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.