Moments of partition functions of 2D Gaussian polymers in the weak disorder regime – II

Abstract

Let $W_N(\mathit{\beta })={\mathrm{E}}_0\left[e^{{\sum }_{n=1}^N\mathit{\beta }\mathit{\omega }(n,S_n)-N{\mathit{\beta }}^2∕2}\right]$ be the partition function of a two-dimensional directed polymer in a random environment, where $\mathit{\omega }(i,x),i\in \mathbb{N},x\in {\mathbb{Z}}^2$ are i.i.d. standard normal and $\{S_n\}$ is the path of a simple random walk. With $\mathit{\beta }={\mathit{\beta }}_N=\hat{\mathit{\beta }}\sqrt{\mathrm{\pi }∕logN}$ and $\hat{\mathit{\beta }}\in (0,1)$ (the subcritical window), $log{W}_N({\mathit{\beta }}_N)$ is known to converge in distribution to a Gaussian law of mean $-{\mathit{\lambda }}^2∕2$ and variance ${\mathit{\lambda }}^2$, with ${\mathit{\lambda }}^2=log(1∕(1-{\hat{\mathit{\beta }}}^2))$ (Caravenna, Sun, Zygouras, Ann. Appl. Probab. (2017)). We study in this paper the moments $\mathbb{E}[W_N{({\mathit{\beta }}_N)}^q]$ in the subcritical window, and prove a lower bound that matches to leading order, for $q=O(\sqrt{logN})$, the upper bound derived by us in Cosco, Zeitouni, Comm. Math. Phys. (2023). The analysis is based on appropriate decouplings and a Poisson convergence that uses the method of “two moments suffice”.