Moments of partition functions of 2D Gaussian polymers in the weak disorder regime -- I

Let $W_N(\beta) = \mathrm{E}_0\left[e^{ \sum_{n=1}^N \beta\omega(n,S_n) - N\beta^2/2}\right]$ be the partition function of a two-dimensional directed polymer in a random environment, where $\omega(i,x), i\in \mathbb{Z}_+, x\in \mathbb{Z}^2$ are i.i.d.\ standard normal and $\{S_n\}$ is the path of a random walk. With $\beta=\beta_N=\hat\beta \sqrt{\pi/\log N}$ and $\hat \beta\in (0,1)$ (the subcritical window), $\log W_N(\beta_N)$ is known to converge in distribution to a Gaussian law of mean $-\lambda^2/2$ and variance $\lambda^2$, with $\lambda^2=\log \big(1/(1-\hat\beta^2\big)$ (Caravenna, Sun, Zygouras, Ann. Appl. Probab. (2017)). We study in this paper the moments $\mathbb{E} [W_N( \beta_N)^q]$ in the subcritical window, for $q=O(\sqrt{\log N})$. The analysis is based on ruling out triple intersections
Comment: 24 pages. V1 corrected a typo and added an acknowledgement. V2 significantly strengthens the main result by getting rid of the "nuisance term" from previous versions. V3 is the revised version. It corrects some typos, fixes an issue with the definition of the function phi in Section 3.5 and adds Lemma 3.8 to describe properties of phi. The upper bound in Theorem 2.1 is now slightly more precise