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 ∼t. Our approach consists in a study of the generating function GΔx(λ)=-nλnpn(Δx) for the probabilities pn(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/Δx2, where f(λ)-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.