# Graduate Student / Postdoc Seminar

#### Directed Polymers in Random Environment with Heavy Tails

Speaker: Oren Louidor

Location: Warren Weaver Hall 1302

Date: Friday, February 26, 2010, 1 p.m.

Synopsis:

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)