Particle-number distribution in large fluctuations at the tip of branching random walks

TL;DR

This paper studies the distribution of particles near the tip of branching random walks during large fluctuations, revealing a nontrivial scaling law for the generating function and growth of particle numbers.

Contribution

It introduces a new scaling law for the generating function of particle counts near the tip of branching random walks, connecting it to FKPP equation solutions.

Findings

Mean particle number grows exponentially with interval size.

The generating function follows a complex scaling law involving a combined variable.

Growth of typical particle number is slower than exponential, with subleading corrections.

Abstract

We investigate properties of the particle distribution near the tip of one-dimensional branching random walks at large times $t$, focusing on unusual realizations in which the rightmost lead particle is very far ahead of its expected position - but still within a distance smaller than the diffusion radius $\sim\sqrt{t}$. Our approach consists in a study of the generating function $G_{\Delta x}(\lambda)=\sum_n \lambda^n p_n(\Delta x)$ for the probabilities $p_n(\Delta x)$ of observing $n$ particles in an interval of given size $\Delta x$ from the lead particle to its left, fixing the position of the latter. This generating function can be expressed with the help of functions solving the Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation with suitable initial conditions. In the infinite-time and large-$\Delta x$ limits, we find that the mean number of particles in the interval grows exponentially with $\Delta x$, and that the generating function obeys a nontrivial scaling law, depending on $\Delta x$ and $\lambda$ through the combined variable $[\Delta x-f(\lambda)]^{3}/\Delta x^2$, where $f(\lambda)\equiv -\ln(1-\lambda)-\ln[-\ln(1-\lambda)]$. From this property, one may conjecture that the growth of the typical particle number with the size of the interval is slower than exponential, but, surprisingly enough, only by a subleading factor at large $\Delta x$. The scaling we argue is consistent with results from a numerical integration of the FKPP equation.