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We study the model of Directed Polymers in Random Environment in 1+1dimensions, where the distribution at a site has a tail which decays regularlypolynomially with power \alpha, where \alpha \in (0,2). After proper scaling oftemperature \beta^{-1}, we show strong localization of the polymer to afavorable region in the environment where energy and entropy are best balanced.We prove that this region has a weak limit under linear scaling and identifythe limiting distribution as an (\alpha, \beta)-indexed family of measures onLipschitz curves lying inside the 45-degrees-rotated square with unit diagonal.In particular, this shows order n transversal fluctuations of the polymer. If,and only if, \alpha is small enough, we find that there exists a randomcritical temperature below which, but not above, the effect of the environmentis macroscopic. The results carry over to d+1 dimensions for d>1 with minormodifications.

Directed polymers in random environment with heavy tails