Anomalous scaling law for the two-dimensional Gaussian free field

We consider level-set percolation for the Gaussian free field $φ$ on $\mathbb{Z}^2$ regularised by a mass of order $N^{-2}$, and investigate the probability $θ_N(a)$ that the cluster of the origin in $\{φ\geq \sqrt{g_N} a\}$ reaches distance $N$ while remaining finite, where $g_N$ is the variance of