Self-Attractive Random Walks: The Case of Critical Drifts
  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. 
Dmitry Ioffe and Yvan Velenik
math.PR cond-mat.stat-mech math-ph math.MP