Comptes Rendus
Probability Theory
Limiting laws of supercritical branching random walks
Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 535-538.

In this Note, we make explicit the limit law of the renormalized supercritical branching random walk, giving credit to a conjecture formulated in Barral et al. (2012) [5] for a continuous analogue 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 (Derrida and Spohn, 1988 [9]) about the Poisson–Dirichlet nature of the jumps in the limit, and precising the conjecture by giving the spatial distribution of these jumps.

Cette Note explicite la loi limite dʼun processus de branchement supercritique renormalisé, confortant ainsi une conjecture formulée dans Barral et al. (2012) [5] pour un analogue continu de cette marche. Dans le cas dʼune marche aléatoire de branchement sur un arbre homogène, nous donnons la loi limite de la mesure de Gibbs renormalisée associée, confirmant pour ce modèle discret des conjectures formulées par des physiciens (Derrida et Spohn, 1988 [9]) à propos de la nature Poisson–Dirichlet des sauts observés à la limite, tout en donnant la distribution spatiale de ces sauts.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.05.013
Julien Barral 1; Rémi Rhodes 2; Vincent Vargas 2

1 LAGA (UMR 7539), département de mathématiques, institut Galilée, université Paris 13, 99, avenue Jean-Baptiste-Clément, 93430 Villetaneuse, France
2 Ceremade (UMR 7534), université Paris-Dauphine, place du maréchal-de-Lattre-de-Tassigny, 75775 Paris cedex 16, France
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Julien Barral; Rémi Rhodes; Vincent Vargas. Limiting laws of supercritical branching random walks. Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 535-538. doi : 10.1016/j.crma.2012.05.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.05.013/

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