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

We consider a transient random walk (X-n) 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 X-n is of order of magnitude n(Lambda), with Lambda is an element of (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).