Almost-sure central limit theorem for directed polymers and random corrections

We consider a general model of directed polymers on the lattice Z^d, d≥3, weakly coupled to a random environment. We prove that the central limit theorem holds almost surely for the discrete time random walk X_T associated to the polymer. Moreover we show that the random corrections to the cumulants of X_T are finite, starting from some dimension depending on the index of the cumulants, and that there are corresponding random corrections of order T−k=2, k = 1;2; : : :, in the asymptotic expansion of the expectations of smooth functions of X_T . We finally prove a kind of local theorem showing that the ratio of the probabilities of the events X_t = y to the corresponding probabilities with no randomness, in the region of “moderate” deviations from the average drift bT , are, for almost all choices of the environment, uniformly close, as T goes to infinity, to a functional of the environment ”as seen from (T; y)”.