Diffusivity of Rescaled Random Polymer in Random Environment in dimensions 1 and 2

We show random polymer is diffusive in dimensions 1 and 2 in probability in an intermediate scaling regime. The scale is $\beta =o(N^{-1\mathrm{/}4})$ in d=1 and $\beta =o((\log N{)}^{-1\mathrm{/}2})$ in $d=2$ as $N\to \mathrm{\infty }$.