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Comptes Rendus Mathématique
Volume 350, n° 9-10
pages 535-538 (mai 2012)
Doi : 10.1016/j.crma.2012.05.013
Received : 28 Mars 2012 ;  accepted : 29 May 2012
Limiting laws of supercritical branching random walks
Lois limites de marches aléatoires de branchement supercritiques
 

Julien Barral a , Rémi Rhodes b , Vincent Vargas b
a LAGA (UMR 7539), département de mathématiques, institut Galilée, université Paris 13, 99, avenue Jean-Baptiste-Clément, 93430 Villetaneuse, France 
b Ceremade (UMR 7534), université Paris-Dauphine, place du maréchal-de-Lattre-de-Tassigny, 75775 Paris cedex 16, France 

Abstract

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.

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Résumé

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.

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© 2012  Published by Elsevier Masson SAS de la part de Académie des sciences.
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