title: Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion.
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 X_{1}(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=X_{1}(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[-(c^{2}/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.
authors.affiliation: Collège de France, 11 Place Marcelin Berthelot, 75005 Paris, France and Laboratoire de Physique Statistique, École Normale Supérieure, 24 Rue Lhomond, 75005 Paris, France.
authors.forename: Bernard
authors.initials: B
authors.lastname: Derrida
authors.affiliation: Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel.
authors.forename: Baruch
authors.initials: B
authors.lastname: Meerson
authors.affiliation: Keldysh Institute of Applied Mathematics, Moscow 125047, Russia.
authors.forename: Pavel V
authors.initials: PV
authors.lastname: Sasorov