On the Long-range Directed Polymer Model

TL;DR

This paper investigates the long-range directed polymer model in a random environment, revealing phase transitions, invariance principles, and super-stable motion behaviors, extending classical results to models with long-range interactions.

Contribution

It extends key results of the nearest-neighbor directed polymer model to long-range models on bZ, including phase transition analysis and behavior characterization in different disorder regimes.

Findings

In the weak disorder regime, the polymer satisfies an invariance principle.

In the very strong disorder regime, the endpoint distribution has macroscopic atoms.

For eta>0 and lpha, the model exhibits super-lpha-stable motion.

Abstract

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.