Anomalous scaling law for the two-dimensional Gaussian free field

TL;DR

This paper establishes a sharp scaling law for the probability of large clusters in the Gaussian free field on a 2D lattice, revealing a transition from logarithmic to polynomial decay across a critical window.

Contribution

It introduces a precise scaling law for cluster probabilities in 2D Gaussian free fields, identifying a transition across a correlation length scale with an explicit decay exponent.

Findings

Probability transitions from fractional logarithmic to polynomial decay.

Explicit decay exponent  identified in the scaling law.

Contrasts with 3D cases where regular polynomial decay occurs.

Abstract

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.