Phase transition for the Once-reinforced random walk on Zd-like trees

In this short paper, we consider the Once-reinforced random walk with reinforcement parameter a on trees with bounded degree which are transient for the simple random walk. On each of these trees, we prove that there exists an explicit critical parameter a0 such that the Once-reinforced random walk is almost surely recurrent if a>a0 and almost surely transient if a<a0. This provides the first examples of phase transition for the Once-reinforced random walk.