Phase transition for the Once-reinforced random walk on ${\mathbb{Z}}^d$-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 $a_0$ such that the Once-reinforced random walk is almost surely recurrent if $a>a_0$ and almost surely transient if $a<a_0$. This provides the first examples of phase transition for the Once-reinforced random walk.