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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+-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.
Shuo Qin (Mon,) studied this question.