Probabilistic picture for particle number densities in stretched tips of the branching Brownian motion
arXiv:2207.07672 · doi:10.1209/0295-5075/aca699
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 , that we take very large, when this extreme particle is conditioned to arrive at a predefined position chosen far ahead of its expected position . We recover the previously-conjectured fact that the typical number density of particles a distance to the left of the lead particle, when both and are large, is smaller than the mean number density by a factor proportional to , where 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 may be inferred.
arXiv comment7 pages, 1 figure. v2: significant improvements to the text, numerous clarifications made. Approach and results unchanged. Version accepted for publications in EPL
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