Abstract: Self-attractive random walks (polymers) undergo a phase transition in terms of the applied drift (force): 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 d ≥ 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.
Among other things, it is proved in REF that the walk is ballistic at criticality.
Self-Attractive Random Walks: The Case of Critical Drifts
Commun. Math. Phys. 313, 209-235 (2012)
Mathematics Physics