Abstract

The directed polymer was introduced by Huse and Henley as a model for the domain wall in a ferromagnetic Ising model with random bond impurities. This model depends on a parameter $\beta$, the inverse temperature. We consider the intermediate disorder regime, which consists in taking $\beta$ to depend on the length of the polymer 2n, with $\beta=n^{$ for some $\alpha>0$. In this regime, there is a critical phase transition that happens at $\alpha=1/4$. When $\alpha > 1/4$, the fluctuations of the free energy are of order $n}{(1-4\alpha)/4}$ and converge to a Gaussian. For $\alpha < 1/4$, it was conjectured that the polymer should fall back in the Kardar—Parisi—Zhang universality class, and that the fluctuations should instead be of order $n^{(1-4\alpha)/3}$, and converge after rescaling to the Tracy—Widom GUE distribution. In this talk, I will sketch a proof of this conjecture for $1/8 < \alpha < 1/4$ for arbitrary i.i.d weights with exponential moments, using a kind of “local chaos expansion”.