Directed polymers in a random environment with heavy tails

We study the model of directed polymers in a random environment in 1 + 1 dimensions, where the distribution at a site has a tail that decays regularly polynomially with power α, where α ε (0,2). After proper scaling of temperature β^-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 (α, β)-indexed family of measures on Lipschitz curves lying inside the 45°-rotated square with unit diagonal. In particular, this shows order-n transversal fluctuations of the polymer. If, and only if, α is small enough, we find that there exists a random critical temperature below which, but not above which, the effect of the environment is macroscopic. The results carry over to d + 1 dimensions for d > 1 with minor modifications.