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

Abstract

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 $\omega (i,x),i\in \mathbb{N},x\in {\mathbb{Z}}^2$ 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 ${\lambda }^2=\log ((1-{\hat{\beta }}^2{)}^{-1})$ (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, and prove a lower bound that matches for $q=O(\sqrt{\log N})$ the upper bound derived by us in Cosco, Zeitouni, arXiv:2112.03767 [math.PR]. The analysis is based on appropriate decouplings and a Poisson convergence that uses the method of ''two moments suffice''.

26 pages