Phenomenological picture of fluctuations in branching random walks

Abstract

We propose a picture of the fluctuations in branching random walks, which leads to predictions for the distribution of a random variable that characterizes the position of the bulk of the particles. We also interpret the $1/\sqrt{t}$ correction to the average position of the rightmost particle of a branching random walk for large times $t\gg 1$, computed by Ebert and Van Saarloos, as fluctuations on top of the mean-field approximation of this process with a Brunet-Derrida cutoff at the tip that simulates discreteness. Our analytical formulas successfully compare to numerical simulations of a particular model of branching random walk.

32 pages, 6 figures. v2: one Appendix added to provide more calculation details, misprints corrected, figure layout improved. To appear in Phys.Rev.E

References in corpus (1)

• Quantum chromodynamics at high energy and statistical physics

Cited by in corpus (13)

• On parton number fluctuations at various stages of the rapidity evolution
• Analytical asymptotics for hard diffraction
• On the genealogy of branching random walks and of directed polymers
• Lecture notes on "Quantum chromodynamics and statistical physics"
• Fluctuations of a swarm of Brownian bees
• Nuclear scattering configurations of onia in different frames
• Persistent fluctuations of the swarm size of Brownian bees
• Probabilistic picture for particle number densities in stretched tips of the branching Brownian motion
• Cross-overs of Bramson's shift at the transition between pulled and pushed fronts
• Probabilities of moderately atypical fluctuations of the size of a swarm of Brownian Bees
• Statistical properties of partonic configurations and diffractive dissociation in high-energy electron-nucleus scattering
• Large deviations for the rightmost position in a branching Brownian motion
• 1-stable fluctuations in branching Brownian motion at critical temperature II: general functionals