Range and critical generations of a random walk on Galton-Watson trees
Résumé
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 $R_n$.
Domaines
Probabilités [math.PR]
Origine : Fichiers produits par l'(les) auteur(s)