Transient random walks in random environment on a Galton-Watson tree

Abstract

We consider a transient random walk (Xn ) in random environment on a Galton-Watson tree. Under fairly general assumptions, we give a sharp and explicit criterion for the asymptotic speed to be positive. As a consequence, situations with zero speed are revealed to occur. In such cases, we prove that Xn is of order of magnitude nΛ, with Λ ∈ (0, 1). We also show that the linearly edge reinforced random walk on a regular tree always has a positive asymptotic speed, which improves a recent result of Collevecchio (Probab Theory Related 136(1):81-101, 2006). © 2008 Springer-Verlag.