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

摘要

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 Λ, with L Î (0,1){Lambda in (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).