  We consider the statistical mechanics of a random polymer with random walks
and disorders in $\mathbb{Z}^d$. The walk collects random disorders along the
way and gets nothing if it visits the same site twice. In the continuum and
weak disorder regime, the partition function as a random variable converges
weakly to a Wiener Chaos expansion when the dimension is lower than the
critical dimension, which is four. A finite temperature case in one dimension
is also discussed. The last case suggests that the end-point behavior of the
polymer is $t^{2/3}$.
