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.

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