New characterization of the weak disorder phase of directed polymers in bounded random environments

TL;DR

This paper characterizes the weak disorder phase of directed polymers in bounded random environments through the integrability of the supremum of an associated martingale, establishing $L^p$ boundedness across this phase.

Contribution

It introduces a new criterion based on the supremum's integrability for identifying the weak disorder phase, extending to a broader class of non-negative martingales with product structures.

Findings

Weak disorder phase characterized by supremum integrability.

Proves $L^p$-boundedness of the martingale in the entire weak disorder phase.

Generalizes the argument to non-negative martingales with product structure.

Abstract

We show that the weak disorder phase for the directed polymer model in a bounded random environment is characterized by the integrability of the running supremum $\sup_{n\in \mathbb N}W_n^\beta$ of the associated martingale $(W_n^\beta)_{n\in \mathbb N}$. Using this characterization, we prove that $(W_n^\beta)_{n\in \mathbb N}$ is $L^p$-bounded in the whole weak disorder phase, for some $p>1$. The argument generalizes to non-negative martingales with a certain product structure.