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In this note, we make explicit the law of the renormalized supercriticalbranching random walk, giving credit to a conjecture formulated in a previousworks 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 expressthe law of the corresponding limiting renormalized Gibbs measures, confirming,in this discrete model, conjectures formulated by physicists about thePoisson-Dirichlet nature of the jumps in the limit, and precising theconjecture by giving the spatial distribution of these jumps.

Limiting laws of supercritical branching random walks