  We study the limiting distribution of particles at the frontier of a
branching random walk. The positions of these particles can be viewed as the
lowest energies of a directed polymer in a random medium in the mean-field
case. We show that the average distances between these leading particles can be
computed as the delay of a traveling wave evolving according to the Fisher-KPP
front equation. These average distances exhibit universal behaviors, different
from those of the probability cascades studied recently in the context of mean
field spin-glasses.
