  In this paper, we study a model of directed polymers in random environment,
where the environment region is a dynamical tube with its width growing
polynomially along with time. It can be viewed as an interpolation of the
pinning model and the classic directed polymer model. We prove weak coupling
limits for the directed polymers on tube in all dimensions, as the inverse
temperature vanishes in a suitable rate. Phase transitions among disorder
irrelevance, marginal relevance and disorder relevance are observed.
