Directed Polymers in Random Environment: Path Localization and Strong Disorder

Abstract We consider directed polymers in random environment. Under mild assumptions on the environment, we prove here: (i) equivalence of decay rate of the partition function with some natural localization properties of the path, (ii) quantitative estimates of the decay of the partition function in dimensions one or two, or at sufficiently low temperature, (iii) existence of quenched free energy. In particular, we generalize to general environments, some of the results recently obtained by P. Carmona and Y. Hu for a Gaussian environment. We do not discuss here superdiffusivity or critical exponents.