Fractal dimension of critical Gaussian free field sign clusters

Abstract

We study metric graph sign clusters of the Gaussian free field on a large class of transient weighted graphs $G$ that are sufficiently low-dimensional, i.e. below the mean-field regime, and prove a macroscopic equivalence property: several natural ways to define macroscopic clusters - involving any of diameter, volume, or capacity functionals - are in fact the same. As a corollary, macroscopic clusters are characterized in low dimensions as those containing large loops in the corresponding loop soup picture, in contrast with the typical behavior in high dimensions. We then apply these results to the small-mesh scaling limit of critical clusters on $\varepsilon\mathbb{Z}^d$, $d=3,4,5$, as $\varepsilon \downarrow 0$. We prove that every subsequential scaling limit consists of sets with positive Brownian capacity whose Hausdorff and Minkowski dimensions are both equal to $1+\tfrac d2$, as conjectured by Werner. We also derive two-point crossing estimates in the limit, and show that each limiting loop soup cluster is the closure of the union of the scaling limits of the macroscopic loops it contains. Remarkably, unlike with planar Bernoulli percolation for instance, the determination of the Hausdorff dimension does not emerge from direct calculations involving a limiting object (such as $\text{SLE}_6$), but rather from the distinctive features of the discrete model itself.