TOPICS ON THE LONG-RANGE DIRECTED POLYMER MODEL

Abstract

We study a long-range directed polymer model in a random environment, where the underlying random walk lies in the domain of attraction of an α-stable Lévy process for some α ∈ (0, 2]. As the inverse temperature β increases, the model undergoes a phase transition from weak disorder to very strong disorder. In this thesis, we extend most of the important results known for the classic nearest-neighbor directed polymer model on Z^{1+d} to the long-range model on Z^{1+1}. More precisely, we give a criterion to test the existence of the weak disorder regime. For α ∈ (1, 2] and some special cases for α=1, we show that the model is in the very strong disorder regime whenever β > 0 by giving explicit upper bounds on the free energy. If strong disorder holds for the model as long as β > 0, then we can also provide a lower bound for the free energy. Furthermore, we show that in the entire weak disorder regime, the polymer satisfies an analogue of the invariance principle, while in the so-called very strong disorder regime, the polymer end-point distribution contains macroscopic atoms and under some extra mild conditions, the polymer has a super-α-stable motion.