Genealogy of the extremal process of the branching random walk

TL;DR

This paper investigates the joint convergence of the extremal process of a branching random walk and its genealogical structure, revealing detailed properties of the limiting decorated Poisson process and supercritical Gibbs measures.

Contribution

It extends previous work by analyzing the genealogical information alongside the extremal process, providing new insights into the decoration law and Gibbs measures in the limit.

Findings

Joint convergence of extremal process and genealogy established.

Characterization of the decoration law in the limiting process.

Analysis of supercritical Gibbs measures in the branching random walk.

Abstract

The extremal process of a branching random walk is the point measure recording the position of particles alive at time $n$, shifted around the expected position of the minimal position. Madaule proved that this point measure converges, as $n \to \infty$, toward a randomly shifted, decorated Poisson point process. In this article, we study the joint convergence of the extremal process together with its genealogical informations. This result is then used to describe the law of the decoration in the limiting process, as well as to study the supercritical Gibbs measures of the branching random walk.