Gaussian fluctuations for the directed polymer partition function in dimension d≥3 and in the whole L2-region

We consider the discrete directed polymer model with i.i.d. environment and we study the fluctuations 8 of the tail n(W∞ − Wn) of the normalized partition function. It was proven by Comets and Liu [18], that 9 for sufficiently high temperature, the fluctuations converge in distribution towards the product of the limiting par10 tition function and an independent Gaussian random variable. We extend the result to the whole L-region, which 11 is predicted to be the maximal high-temperature region where the Gaussian fluctuations should occur under the 12 considered scaling. To do so, we manage to avoid the heavy 4th-moment computation and instead rely on the local 13 limit theorem for polymers [47, 49] and homogenization. 14 MSC 2010 subject classifications: Primary 60K37, 60K37; secondary 60F05. 15