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We study the limiting distribution of particles at the frontier of abranching random walk. The positions of these particles can be viewed as thelowest energies of a directed polymer in a random medium in the mean-fieldcase. We show that the average distances between these leading particles can becomputed as the delay of a traveling wave evolving according to the Fisher-KPPfront equation. These average distances exhibit universal behaviors, differentfrom those of the probability cascades studied recently in the context of meanfield spin-glasses.

Statistics at the tip of a branching random walk and the delay of traveling waves