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 at a distance Δ to the left of the lead particle, when both Δ and are large, is smaller than the mean number density by a factor proportional to , where ζ 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 ζ may be inferred.