Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Range and critical generations of a random walk on Galton–Watson trees

Pierre Andreoletti and Xinxin Chen

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Abstract

In this paper we consider a random walk in random environment on a tree and focus on the boundary case for the underlying branching potential. We study the range $R_{n}$ of this walk up to time $n$ and obtain its correct asymptotic in probability which is of order $n/\log n$. This result is a consequence of the asymptotical behavior of the number of visited sites at generations of order $(\log n)^{2}$, which turn out to be the most visited generations. Our proof which involves a quenched analysis gives a description of the typical environments responsible for the behavior of $R_{n}$.

Résumé

Dans cet article nous considérons une marche aléatoire en milieu aléatoire sur un arbre, en nous concentrant sur le cas frontière du potentiel branchant sous-jacent. Nous étudions le nombre de points visités par cette marche avant l’instant $n$, $R_{n}$, et obtenons son comportement asymptotique en probabilité qui est de l’ordre de $n/\log n$. Ce résultat est une conséquence du comportement asymptotique du nombre de points visités par la marche au niveau des générations critiques, c’est à dire en $(\log n)^{2}$. La preuve permet une description des environnements typiques qui conduisent au comportement de $R_{n}$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 1 (2018), 466-513.

Dates
Received: 14 October 2015
Revised: 24 October 2016
Accepted: 28 November 2016
First available in Project Euclid: 19 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1519030836

Digital Object Identifier
doi:10.1214/16-AIHP812

Mathematical Reviews number (MathSciNet)
MR3765897

Zentralblatt MATH identifier
06880062

Subjects
Primary: 60K37: Processes in random environments 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G50: Sums of independent random variables; random walks

Keywords
Random walks Range Random environment Branching random walk

Citation

Andreoletti, Pierre; Chen, Xinxin. Range and critical generations of a random walk on Galton–Watson trees. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 1, 466--513. doi:10.1214/16-AIHP812. https://projecteuclid.org/euclid.aihp/1519030836


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