Weak-disorder moment conjecture for two-dimensional Gaussian polymers

Let $W_N$ be the normalized partition function of the two-dimensional Gaussian polymer, let $\lambda(\hat\beta)$ denote the model-dependent parameter appearing in the moment estimates, and define

$$q_\star^2(N,\hat\beta)=\frac{2}{\lambda^2(\hat\beta)}\log N.$$

Weak-disorder moment conjecture. For all $\delta\in(0,1)$ and $\hat\beta\in(0,1)$, uniformly for all $q<(1-\delta)q_\star(N,\hat\beta)$, as $N\to\infty$,

$$\mathbb{E}\left[W_N^q\right]\leq \exp\left(\lambda^2\frac{q(q-1)}{2}(1+o(1))\right).$$

This conjecture predicts the optimal Gaussian-type upper bound for moments of the normalized partition function throughout the subcritical range below the threshold $q_\star(N,\hat\beta)$. The surrounding discussion motivates it as the estimate needed to capture the expected behavior of maxima on spatial scales up to order $N^{1/2}$; the statement is presented as a conjecture, and no resolution is given here.