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

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 $G_{Δx}(λ)=\sum_n λ^n p_n(Δx)$ for the probabilities $p_n(Δx)$ of observing $n$ particles in an interval of given size $Δx$ 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-$Δx$ limits, we find that the mean number of particles in the interval grows exponentially with $Δx$, and that the generating function obeys a nontrivial scaling law, depending on $Δx$ and $λ$ through the combined variable $[Δx-f(λ)]^{3}/Δx^2$, where $f(λ)\equiv -\ln(1-λ)-\ln[-\ln(1-λ)]$. 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 $Δx$. The scaling we argue is consistent with results from a numerical integration of the FKPP equation.