Positive and negative moments for directed polymers in random environment in weak disorder

TL;DR

This paper characterizes the weak disorder phase of directed polymers in random environments, removing the previous boundedness assumption and establishing new moment bounds for the polymer martingale.

Contribution

It extends the understanding of the weak disorder phase by proving moment bounds without the boundedness condition on the environments.

Findings

Weak disorder phase implies boundedness of the polymer martingale in certain $L^p$ spaces.

Removal of the boundedness assumption on the environments.

Establishment of moment bounds for the polymer martingale in the weak disorder phase.

Abstract

Very recently, Junk [11] showed that for directed polymers in bounded random environments, the weak disorder (uniform integrable) phase implies that the polymer martingale is bounded in $L^p$ for some $p>1$ and also in $L^q$ for some $q<0$. Here, we establish this characterization of the weak disorder phase without requiring the boundedness assumption on the environments.