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We consider the discrete directed polymer model with i.i.d. environment andwe study the fluctuations of the tail $n^{(d-2)/4}(W_\infty - W_n)$ of thenormalized partition function. It was proven by Comets and Liu, that forsufficiently high temperature, the fluctuations converge in distributiontowards the product of the limiting partition function and an independentGaussian random variable. We extend the result to the whole $L^2$-region, whichis predicted to be the maximal high-temperature region where the Gaussianfluctuations should occur under the considered scaling. To do so, we manage toavoid the heavy 4th-moment computation and instead rely on the local limittheorem for polymers and homogenization.

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