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We show that the weak disorder phase for the directed polymer model in abounded random environment is characterized by the integrability of the runningsupremum $\sup_{n\in \mathbb N}W_n^\beta$ 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 disorderphase, for some $p>1$. The argument generalizes to non-negative martingaleswith a certain product structure.

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