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

Pierre Andreoletti (1) , Xinxin Chen (2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 PSPM - Probabilités, statistique, physique mathématique

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 $R_n$.

Mots clés: branching random walk; range; random environment; random walks

Domaines: Probabilités [math.PR]

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, 2018, 54 (1). ⟨hal-01211586v2⟩