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2022 Scaling limits of tree-valued branching random walks
Thomas Duquesne, Robin Khanfir, Shen Lin, Niccolò Torri
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Electron. J. Probab. 27: 1-54 (2022). DOI: 10.1214/22-EJP741

Abstract

We consider a branching random walk (BRW) taking its values in the b-ary rooted tree Wb (i.e. the set of finite words written in the alphabet {1,,b}, with b2). The BRW is indexed by a critical Galton–Watson tree conditioned to have n vertices; its offspring distribution is aperiodic and is in the domain of attraction of a γ-stable law, γ(1,2]. The jumps of the BRW are those of a nearest-neighbour null-recurrent random walk on Wb (reflection at the root of Wb and otherwise: probability 12 to move closer to the root of Wb and probability 1(2b) to move away from it to one of the b sites above). We denote by Rb(n) the range of the BRW in Wb which is the set of all sites in Wb visited by the BRW. We first prove a law of large numbers for #Rb(n) and we also prove that if we equip Rb(n) (which is a random subtree of Wb) with its graph-distance dgr, then there exists a scaling sequence (an)nN satisfying an such that the metric space (Rb(n),an1dgr), equipped with its normalised empirical measure, converges to the reflected Brownian cactus with γ-stable branching mechanism: namely, a random compact real tree that is a variant of the Brownian cactus introduced by N. Curien, J-F. Le Gall and G. Miermont in [7].

Funding Statement

N. Torri was supported by the project Labex MME-DII (ANR11-LBX-0023-01).

Acknowledgments

The authors would like to thank the referee and the editor for their helpful suggestions.

Citation

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Thomas Duquesne. Robin Khanfir. Shen Lin. Niccolò Torri. "Scaling limits of tree-valued branching random walks." Electron. J. Probab. 27 1 - 54, 2022. https://doi.org/10.1214/22-EJP741

Information

Received: 15 April 2021; Accepted: 8 January 2022; Published: 2022
First available in Project Euclid: 28 January 2022

Digital Object Identifier: 10.1214/22-EJP741

Subjects:
Primary: 60F17 , 60G50 , 60G52 , 60J80

Keywords: branching random walks , Brownian cactus , Brownian snake , Galton–Watson tree , real tree , Scaling limit , Superprocess

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