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Condensed Matter > Statistical Mechanics
Title: Probabilistic picture for particle number densities in stretched tips of the branching Brownian motion
(Submitted on 15 Jul 2022 (v1), last revised 29 Nov 2022 (this version, v2))
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.
Submission history
From: Stephane Munier [view email][v1] Fri, 15 Jul 2022 18:00:05 GMT (76kb,D)
[v2] Tue, 29 Nov 2022 06:50:00 GMT (78kb,D)
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