February 26: Oren Louidor, CIMS
Directed polymers in random environment with heavy tails
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^{1}, 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.
(With Antonio Auffinger)
