Gaussian fluctuations for the directed polymer partition function for $d\ge 3$ and in the whole $L^2$-region

We consider the discrete directed polymer model with i.i.d. environment and we study the fluctuations of the tail $n^{(d-2)\mathrm{/}4}(W_{\mathrm{\infty }}-W_n)$ of the normalized partition function. It was proven by Comets and Liu, that for sufficiently high temperature, the fluctuations converge in distribution towards the product of the limiting partition function and an independent Gaussian random variable. We extend the result to the whole $L^2$-region, which is predicted to be the maximal high-temperature region where the Gaussian fluctuations should occur under the considered scaling. To do so, we manage to avoid the heavy 4th-moment computation and instead rely on the local limit theorem for polymers and homogenization.