  Self-attractive random walks undergo a phase transition in terms of the
applied 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 is
sub-ballistic. We show that, in any dimension at least 2, this transition is of
first order. In fact, we prove that the walk is already ballistic at critical
drifts, and establish the corresponding LLN and CLT.
