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

Abstract

Let $W_N(β) = \mathrm{E}_0\left[e^{ \sum_{n=1}^N βω(n,S_n) - Nβ^2/2}\right]$ be the partition function of a two-dimensional directed polymer in a random environment, where $ω(i,x), i\in \mathbb{Z}_+, x\in \mathbb{Z}^2$ are i.i.d.\ standard normal and $\{S_n\}$ is the path of a random walk. With $β=β_N=\hatβ\sqrt{π/\log N}$ and $\hat β\in (0,1)$ (the subcritical window), $\log W_N(β_N)$ is known to converge in distribution to a Gaussian law of mean $-λ^2/2$ and variance $λ^2$, with $λ^2=\log \big(1/(1-\hatβ^2\big)$ (Caravenna, Sun, Zygouras, Ann. Appl. Probab. (2017)). We study in this paper the moments $\mathbb{E} [W_N( β_N)^q]$ in the subcritical window, for $q=O(\sqrt{\log N})$. The analysis is based on ruling out triple intersections