A branching random walk seen from the tip

TL;DR

This paper links the statistical properties of the rightmost points in branching Brownian motion to Fisher-KPP traveling wave solutions, revealing a superposability property and extending results to general branching random walks.

Contribution

It demonstrates that all time-dependent statistics of the rightmost points can be derived from Fisher-KPP wave solutions and introduces the superposability property of the limiting measure.

Findings

Distribution of distances has a long time limit as Fisher-KPP wave delay

Limiting measure exhibits superposability property

Results extend to general branching random walks

Abstract

We show that all the time-dependent statistical properties of the rightmost points of a branching Brownian motion can be extracted from the traveling wave solutions of the Fisher-KPP equation. We show that the distribution of all the distances between the rightmost points has a long time limit which can be understood as the delay of the Fisher-KPP traveling waves when the initial condition is modified. The limiting measure exhibits the surprising property of superposability: the statistical properties of the distances between the rightmost points of the union of two realizations of the branching Brownian motion shifted by arbitrary amounts are the same as those of a single realization. We discuss the extension of our results to more general branching random walks.