ALMOST-SURE CENTRAL-LIMIT-THEOREM FOR DIRECTED POLYMERS AND RANDOM CORRECTIONS

We consider a general model of directed polymers on the lattice;Z(nu), nu greater than or equal to 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 s how that the random corrections to the cumulants of X-T are finite, st arting from some dimension depending on the index of the cumulants, an d that there are corresponding random corrections of order T-k/2, k = 1,2,..., in the asymptotic expansion of the expectations of smooth fun ctions of X-T, Full proofs are carried out for the first two cumulants . We finally prove a kind of local theorem showing that the ratio of t he probabilities of the events X-t = y to the corresponding probabilit ies with no randomness, in the region \y - bT\ = o(T2/3) of ''moderate '' deviations from the average drift bT, are, for almost all choices o f the environment, uniformly close, as T --> infinity, to a functional of the environment ''as seen from (T, y)''.