文摘
We study the long-range directed polymer model on \(\mathbbm {Z}\) in a random environment, where the underlying random walk lies in the domain of attraction of an \(\alpha \)-stable process for some \(\alpha \in (0,2]\). Similar to the more classic nearest-neighbor directed polymer model, as the inverse temperature \(\beta \) increases, the model undergoes a transition from a weak disorder regime to a strong disorder regime. We extend most of the important results known for the nearest-neighbor directed polymer model on \(\mathbbm {Z}^d\) to the long-range model on \(\mathbbm {Z}\). More precisely, we show that in the entire weak disorder regime, the polymer satisfies an analogue of invariance principle, while in the so-called very strong disorder regime, the polymer end point distribution contains macroscopic atoms and under some mild conditions, the polymer has a super-\(\alpha \)-stable motion. Furthermore, for \(\alpha \in (1,2]\), we show that the model is in the very strong disorder regime whenever \(\beta >0\), and we give explicit bounds on the free energy.
