Genealogy of the extremal process of the branching random walk

Bastien Mallein

Published 2016-06-06Version 1

Let $(\mathbf{T},V)$ be a branching random walk on the real line. The extremal process of the branching random walk is the point process of the position of particles at time $n$ shifted by the position of the minimum. Madaule proved that the extremal process converges toward a shifted decorated Poisson point process. In this article we study the joint convergence of the extremal process with its genealogy informations. his result is then used to characterize the law of the decoration in the limiting process as well as to study the supercritical Gibbs measures of the branching random walk.