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

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.