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

TL;DR

This paper investigates the distribution of extremal particles in a branching random walk, linking their positions to traveling wave delays in the Fisher-KPP equation, revealing universal behaviors distinct from spin-glass models.

Contribution

It establishes a connection between the frontier distribution of branching random walks and traveling wave delays, providing new insights into their universal properties.

Findings

Average distances between leading particles can be computed via Fisher-KPP wave delays.

The behaviors observed are universal and differ from mean field spin-glass models.

The study offers a new perspective on extremal processes in branching structures.

Abstract

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.