Rate of convergence for polymers in a weak disorder

We consider directed polymers in random environment on the lattice Zd at small inverse temperature and dimension d≥3. Then, the normalized partition function Wn is a regular martingale with limit W. We prove that n(d−2)/4(Wn−W)/Wn converges in distribution to a Gaussian law. Both the polynomial rate of convergence and the scaling with the martingale Wn are different from those for polymers on trees.