Limiting laws of supercritical branching random walks

Julien Barral (1) , Rémi Rhodes (2) , Vincent Vargas (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: branching random walk; supercritical; freezing

Domaines: Probabilités [math.PR]

Julien Barral, Rémi Rhodes, Vincent Vargas. Limiting laws of supercritical branching random walks. Comptes rendus de l'Académie des sciences. Série I, Mathématique, 2012. ⟨hal-00682212v2⟩