Joint localization of directed polymers

We consider (1+1)-dimensional directed polymers in a random potential and provide sufficient conditions guaranteeing joint localization. Joint localization means that for typical realizations of the environment, started at different starting points, all associated endpoint distributions localize common region does not grow with length polymer. In particular, we prove holds when reference walk polymer model is either simple symmetric lattice or Gaussian walk. also very strong disorder property large class space-continuous models, implying usual single