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, and for
polymers started at different starting points, all the associated endpoint
distributions localize in a common random region that does not grow with the
length of the polymer. In particular, we prove that joint localization holds
when the reference random walk of the polymer model is either a simple
symmetric lattice walk or a Gaussian random walk. We also prove that the very
strong disorder property holds for a large class of space-continuous polymer
models, implying the usual single polymer localization.
