Phase Transition Analysis of the Partition Function for Directed Polymers in a Random Environment


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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