  In this paper, we study a model of directed polymers in random environment,
where the environment is restricted to a time-space tube whose spatial width
grows polynomially with time. It can be viewed as an interpolation between the
disordered pinning model and the classic directed polymer model. We prove weak
coupling limits for the directed polymer partition functions in such tube
environments in all dimensions, as the inverse temperature vanishes at a
suitable rate. As the tube width varies, transitions between regimes of
disorder irrelevance, marginal relevance and disorder relevance are observed.
