Phenomenological picture of fluctuations in branching random walks

TL;DR

This paper develops a phenomenological framework for understanding fluctuations in branching random walks, providing analytical predictions for particle distribution and interpreting correction terms as fluctuations beyond mean-field approximations, validated by numerical simulations.

Contribution

It introduces a new phenomenological picture of fluctuations in branching random walks and interprets correction terms as fluctuation effects, supported by analytical formulas and numerical validation.

Findings

Predictions for the distribution of particle positions in branching random walks.

Interpretation of $1/\sqrt{t}$ correction as fluctuation effects.

Analytical formulas match numerical simulations.

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.