Anomalous scaling law for the two-dimensional Gaussian free field

We consider the Gaussian free field $φ$ on $\mathbb{Z}^2$ at large spatial scales $N$ and give sharp bounds on the probability $θ(a,N)$ that the radius of a finite cluster in the excursion set ${φ\geq a}$ on the corresponding metric graph is macroscopic. We prove a scaling law for this probability, by which $θ(a,N)$ 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 $ξ$, which is such that $θ(a,N) \sim θ(0,ξ)(\tfrac{N}ξ)^{-τ}$ for typical heights $a$ as $N/ξ$ diverges, with an explicit exponent $τ$ 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 ${N}/ξ$ away from it.

💡 Research Summary

This paper conducts a rigorous analysis of large-scale connectivity within the two-dimensional Gaussian Free Field (GFF). The central object of study is the probability θ(a,N) that the origin belongs to a finite cluster of the excursion set {φ ≥ a} (the set where the field exceeds a height a) whose diameter is of the order of the system size N. The authors establish the precise asymptotic scaling of this probability, revealing a fundamental departure from the standard scaling hypothesis observed in higher dimensions.

The main result, Theorem 1.1, proves an “anomalous” scaling law. Defining a correlation length ξ(a,N) = N exp(-a² g_N), where g_N ~ (2/π) log N is the field variance, the probability scales as θ(\bar a, N) ~ θ(0, ξ) * (N/ξ)^{-τ} for typical heights a in the off-critical regime (where N/ξ → ∞). The exponent τ is explicitly identified as half the capacity of a unit line segment in R². This is anomalous because the polynomial decay (N/ξ)^{-τ} characterizes the off-critical regime. In stark contrast, at and near the critical point (a=0), the probability exhibits a fractional logarithmic decay, θ(0,N) ~ (log N)^{-1/2}, as previously known. This reverses the pattern seen in three dimensions, where polynomial decay holds at criticality and rapid decay sets in off-criticality.

The proof is a technical tour de force, built on a sophisticated multi-scale analysis. The upper bound proof introduces three carefully chosen scales: ξ « L « M. The smallest scale ξ captures the critical cost θ(0,ξ). The intermediate scale L is chosen large enough to decouple this cost from the rest of the system. The largest scale M is used for a coarse-graining procedure to control the probability of a high-level connection from the boundary of B_L to the boundary of B_N. A key difference from the 3D case is that in 2D, the coarse-graining involves only O(1) boxes when N/ξ is large, leading directly to polynomial decay. The upper bound emerges from a delicate balance between three terms: a local term bounding independent crossing events, and two terms controlling deviations of harmonic averages arising from conditioning.

The lower bound construction mirrors the scale hierarchy from the upper bound and relies crucially on two-dimensional “local uniqueness” properties, which ensure that the optimal connection path is essentially unique.

The mathematical framework heavily utilizes tools from potential theory and random walks. The authors work on the cable system extension of Z², employ the isomorphism theorem linking the GFF to random interlacements, and rely on precise asymptotics for the killed Green’s function. This allows them to translate the problem of high-level crossings in a correlated field into a question about capacities and intersections of random walk paths.

In summary, this work provides a complete and precise description of the scaling behavior for a fundamental connectivity probability in the 2D GFF. It conclusively demonstrates how logarithmic corrections inherent to the two-dimensional critical state fundamentally alter the scaling paradigm, replacing the standard power-law critical decay with a fractional logarithmic decay and shifting the power-law decay to the off-critical regime. The results and the multi-scale methodology developed are likely to influence future studies of critical phenomena in low-dimensional correlated systems.