On the Long-Range Directed Polymer Model

We study the long-range directed polymer model on in a random environment, where the underlying random walk lies in the domain of attraction of an -stable process for some . Similar to the more classic nearest-neighbor directed polymer model, as the inverse temperature 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 to the long-range model on . 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--stable motion. Furthermore, for , we show that the model is in the very strong disorder regime whenever , and we give explicit bounds on the free energy.