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February 2021 On the critical branching random walk III: The critical dimension
Qingsan Zhu
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Ann. Inst. H. Poincaré Probab. Statist. 57(1): 73-93 (February 2021). DOI: 10.1214/20-AIHP1071

Abstract

In this paper, we study the critical branching random walk in the critical dimension, four. We provide the asymptotics of the probability of visiting a fixed finite set and the range of the critical branching random walk conditioned on the total number of offspring. We also prove that conditioned on visiting a finite set, the first visiting point converges in distribution, when the starting point tends to infinity.

Dans cet article, nous étudions la marche aléatoire de branchement critique en dimension critique (quatre). Nous déterminons les probabilités asymptotiques de visite d’un ensemble fini fixé, et l’image de la marche aléatoire de branchement critique conditionnée par le nombre total d’individus. Nous montrons également que, conditionnellement à l’événement de visite d’un ensemble fini, le premier point visité converge en loi, lorsque le point de départ tend vers l’infini.

Acknowledgements

The author is partially supported by ISF grant 1207/15 and ERC starting grant 676970. This work was done when the author was a PhD student in the University of British Columbia. The author would like to thank Professor Omer Angel for supervising his research. We are also indebted to two anonymous referees for a number of useful comments.

Citation

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Qingsan Zhu. "On the critical branching random walk III: The critical dimension." Ann. Inst. H. Poincaré Probab. Statist. 57 (1) 73 - 93, February 2021. https://doi.org/10.1214/20-AIHP1071

Information

Received: 4 July 2019; Revised: 6 April 2020; Accepted: 19 May 2020; Published: February 2021
First available in Project Euclid: 12 March 2021

Digital Object Identifier: 10.1214/20-AIHP1071

Subjects:
Primary: 60G50, 60J80

Rights: Copyright © 2021 Association des Publications de l’Institut Henri Poincaré

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Vol.57 • No. 1 • February 2021
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