On the Long-Range Directed Polymer Model

We study the long-range directed polymer model on Z in a random environment, where the underlying random walk lies in the domain of attraction of an α-stable process for some α∈ (0 , 2]. Similar to the more classic nearest-neighbor directed polymer model, as the inverse temperature β 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 Z^d to the long-range model on 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-α-stable motion. Furthermore, for α∈ (1 , 2] , we show that the model is in the very strong disorder regime whenever β> 0 , and we give explicit bounds on the free energy.