Anomalous scaling law for the two-dimensional Gaussian free field

We consider the Gaussian free field $\varphi$ on $\mathbb{Z}^2$ at large spatial scales $N$ and give sharp bounds on the probability $\theta(a,N)$ that the radius of a finite cluster in the excursion set $\{\varphi \geq a\}$ on the corresponding metric graph is macroscopic. We prove a scaling law for this probability, by which $\theta(a,N)$ transitions from fractional logarithmic decay for near-critical parameters $(a,N)$ to polynomial decay in the off-critical regime. The transition occurs across a certain scaling window determined by a correlation length scale $\xi$, which is such that $\theta(a,N) \sim \theta(0,\xi)(\tfrac{N}{\xi})^{-\tau}$ for typical heights $a$ as $N/\xi$ diverges, with an explicit exponent $\tau$ that we identify in the process. This is in stark contrast with recent results from arXiv:2101.02200 and arXiv:2312.10030 in dimension three, where similar observables are shown to follow regular scaling laws, with polynomial decay at and near criticality, and rapid decay in ${N}/\xi$ away from it.