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

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.