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 similar to root t. Our approach consists in a study of the generating function G(Lambda x)(lambda) = Sigma(lambda n)(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) equivalent to - 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.