  In this note, we make explicit the law of the renormalized supercritical
branching random walk, giving credit to a conjecture formulated in a previous
works of the authors for a continuous analog of the branching random walk.
Also, in the case of a branching random walk on a homogeneous tree, we express
the law of the corresponding limiting renormalized Gibbs measures, confirming,
in this discrete model, conjectures formulated by physicists about the
Poisson-Dirichlet nature of the jumps in the limit, and precising the
conjecture by giving the spatial distribution of these jumps.

Probability