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

TL;DR

This paper investigates the moments of the partition function of 2D Gaussian polymers in the weak disorder regime, revealing detailed asymptotic behavior and extending understanding of the model's probabilistic properties.

Contribution

It provides a detailed analysis of the moments of the partition function in the subcritical window, especially for large moments up to order rom prior work on convergence in distribution.

Findings

Moments  the partition function are characterized for large q.

The analysis rules out triple intersections in the path structure.

Results extend understanding of Gaussian polymer behavior in weak disorder.

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 $\omega(i,x), i\in \mathbb{Z}_+, x\in \mathbb{Z}^2$ 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