On the Central Limit Theorem for the log-partition function of 2D directed polymers

Abstract

The log-partition function $\log W_N({\hat{I}}^2)$ of the two-dimensional directed polymer in random environment is known to converge in distribution to a normal distribution when considering temperature in the subcritical regime ${\hat{I}}^2={\hat{I}}_N^2={\hat{\hat{I}}}^2\sqrt{\ddot{I}€/\log N}$, ${\hat{\hat{I}}}^2\in (0,1)$ (Caravenna, Sun, Zygouras, Ann. Appl. Prob. (2017)). In this paper, we present an elementary proof of this result relying on a decoupling argument and the central limit theorem for sums of independent random variables. The argument is inspired by an analogy of the model to branching random walks.

11 pages