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Directed polymers in random environment with heavy tails

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We study the model of Directed Polymers in Random Environment in 1+1 dimensions, where the environment is i.i.d. with a site distribution having a tail that decays regularly polynomially with power -\alpha, where \alpha \in (0,2). After proper scaling of temperature \beta, we show strong localization of the polymer to an optimal region in the environment where energy and entropy are best balanced. We prove that this region has a weak limit under linear scaling and identify the limiting distribution as an (\alpha, \beta)-indexed family of measures on Lipschitz curves lying inside the 45{\circ}-rotated square with unit diagonal. In particular, this shows order of n for the transversal fluctuations of the polymer. If (and only if) \alpha is small enough, we find that there exists a random critical temperature above which the effect of the environment is not macroscopically noticeable. The results carry over to higher dimensions with minor modifications. Joint work with A. Auffinger .

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