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We show rigorously, under mild assumptions on the distribution of the
environment, that there exists a critical region given in terms of the
strength of the environment in which the partition function of the
directed polymer (random walk) in a random environment undergoes a phase
transition. Exact asymptotics are provided for the strongest and
weakest paths. The exact region for which the second moment of
the partition function remains bounded and a region for which all
moments strictly above one are unbounded are also given.
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