  We study the model of Directed Polymers in Random Environment in 1+1
dimensions, where the distribution at a site has a tail which decays regularly
polynomially with power \alpha, where \alpha \in (0,2). After proper scaling of
temperature \beta^{-1}, we show strong localization of the polymer to a
favorable 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-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 random
critical temperature below which, but not above, the effect of the environment
is macroscopic. The results carry over to d+1 dimensions for d>1 with minor
modifications.
