Genealogy of the extremal process of the branching random walk

Abstract : Let (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 [Mad15] proved that this point 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. This 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.