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 $φ_0$. We prove a scaling law for $θ_N(a)$ which is quantitative simultaneously in $a$ and $N$. It asserts that $θ_N(a)=θ_ξ(a_*)(N/ξ)^{-\vartheta+o(1)}$ as $N/ξ\to\infty$, where the correlation length $ξ$ diverges exponentially as $a$ approaches the critical level $a_*$, the critical one-arm probability $θ_ξ(a_*)$ decays fractional-logarithmically, and the exponent $\vartheta$ is explicit and given by a variational problem involving the logarithmic energy. Thus $θ_N(a)$ crosses over from fractional logarithmic decay at and near criticality ($N\lesssimξ$) to polynomial decay away from it ($N\ggξ$), in stark contrast with dimension three and higher, where similar observables exhibit polynomial decay at and near criticality and rapid decay away from it. The proof relies on a coarse-graining scheme capturing a polynomial rate with sharp exponent, and on sharp logarithmic capacity estimates for planar "porous" sets.