Self-Attractive Random Walks: The Case of Critical Drifts

TL;DR

This paper investigates self-attractive random walks with critical drifts, demonstrating a first-order phase transition and proving ballistic behavior at critical points along with associated limit theorems.

Contribution

It establishes that the phase transition in self-attractive random walks is of first order and proves ballisticity at critical drifts in dimensions two and higher.

Findings

The phase transition is of first order.

Walks are ballistic at critical drifts.

LLN and CLT hold at critical points.

Abstract

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.