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

¹ CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris - 91120 Palaiseau, France
² Department of Physics, Columbia University - New York, NY 10027, USA

^(a) E-mail: dung.le@polytechnique.edu
^(b) E-mail: amh@phys.columbia.edu
^(c) E-mail: stephane.munier@polytechnique.edu (corresponding author)

Received: 24 July 2022
Accepted: 28 November 2022

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.