Directed polymers up to the L_2 threshold

We present a new technique for studying a general model of discrete-time directed polymers on the lattice Z^d, d≥ 3, in an i.i.d. random medium in the range of the parameters where the L_2-norm of the partition function converges. The method is based on the study of the analytic properties of a kind of complex generating function. We obtain a simple general proof of the classical results, such as convergence of the partition function and diffusion (Theorems 2.1 and 2.2), which were so far obtained for particular cases, and derive some new ones on convergence of higher moments (Theorem 2.3) and on the rate of divergence of the L_2-norm of the partition function at the critical point.