Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion

Bernard Derrida, Baruch Meerson, Pavel V. Sasorov

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16 Scopus citations

Abstract

Consider a one-dimensional branching Brownian motion and rescale the coordinate and time so that the rates of branching and diffusion are both equal to 1. If X1(t) is the position of the rightmost particle of the branching Brownian motion at time t, the empirical velocity c of this rightmost particle is defined as c=X1(t)/t. Using the Fisher-Kolmogorov-Petrovsky-Piscounov equation, we evaluate the probability distribution P(c,t) of this empirical velocity c in the long-time t limit for c>2. It is already known that, for a single seed particle, P(c,t)∼exp[-(c2/4-1)t] up to a prefactor that can depend on c and t. Here we show how to determine this prefactor. The result can be easily generalized to the case of multiple seed particles and to branching random walks associated with other traveling-wave equations.

Original languageAmerican English
Article number042139
JournalPhysical Review E
Volume93
Issue number4
DOIs
StatePublished - 29 Apr 2016

Bibliographical note

Publisher Copyright:
© 2016 American Physical Society.

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