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

2 PSPM - Probabilités, statistique, physique mathématique
ICJ - Institut Camille Jordan [Villeurbanne]
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 $n/\log n$. Thisresult 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 analysisgives a description of the typical environments responsible for the behavior of $R_n$.
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https://hal.archives-ouvertes.fr/hal-01211586
Contributor : Pierre Andreoletti <>
Submitted on : Tuesday, June 21, 2016 - 9:11:16 PM
Last modification on : Friday, April 23, 2021 - 3:28:04 PM
Long-term archiving on: : Thursday, September 22, 2016 - 4:46:44 PM

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### Identifiers

• HAL Id : hal-01211586, version 2
• ARXIV : 1510.01121

### Citation

Pierre Andreoletti, Xinxin Chen. Range and critical generations of a random walk on Galton-Watson trees. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2018, 54 (1). ⟨hal-01211586v2⟩

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