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.

arXiv comment32 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 (13)

• Front propagation into unstable states
• Shift in the velocity of a front due to a cut-off
• Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts
• Effects of Fluctuations on Propagating Fronts
• Universal behavior of QCD amplitudes at high energy from general tools of statistical physics
• A phenomenological theory giving the full statistics of the position of fluctuating pulled fronts
• A branching random walk seen from the tip
• Statistics at the tip of a branching random walk and the delay of traveling waves
• Extreme Value Statistics and Traveling Fronts: Various Applications
• Microscopic models of traveling wave equations
• Hybrid method for simulating front propagation in reaction-diffusion systems
• Numerical schemes for continuum models of reaction-diffusion systems subject to internal noise
• Quantum chromodynamics at high energy and statistical physics

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