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Self-attractive random walks undergo a phase transition in terms of theapplied drift: If the drift is strong enough, then the walk is ballistic,whereas in the case of small drifts self-attraction wins and the walk issub-ballistic. We show that, in any dimension at least 2, this transition is offirst order. In fact, we prove that the walk is already ballistic at criticaldrifts, and establish the corresponding LLN and CLT.

Self-Attractive Random Walks: The Case of Critical Drifts