A martingale approach to directed polymers in a random environment

The diffusive behavior for a system of directed polymers in a random environment was first rigorously discussed by Imbrie and Spencer, and then by Bolthausen. By means of some basic properties of martingales we extend some results due to Imbrie and Spencer concerning the asymptotic behaviour of the mean square displacement. We also obtain a Wiener process behaviour with probability one for this system. Bolthausen already used some martingale limit theorems to prove a central limit theorem for this system.