Recurrence and transience of step-reinforced random walks

We solve a conjecture raised by Bertoin (personal communication) on the step-reinforced random walk, a non-Markovian process generalizing the elephant random walk. We show that, under a 2+δ-th moment condition, it undergoes a phase transition in dimensions $d=1,2$ between recurrence and transience, and that it is transient for all parameters in dimension d≥ 3.