Separation and cut edge in macroscopic clusters for metric graph Gaussian free fields

Abstract

We prove that for the Gaussian free field (GFF) on the metric graph of $\mathbb{Z}^d$ (for all $d\ge 3$ except the critical dimension $d_c=6$), with uniformly positive probability there exist two distinct sign clusters of diameter at least $cN$ within a box of size $N$ such that their graph distance is less than $N^{-[(d-2)\vee (2d-8)]}$. This phenomenon contrasts sharply with the two-dimensional case, where the distance between two macroscopic clusters is typically on the order of their diameters, following from the basic property of the scaling limit ``conformal loop ensembles'' $\mathrm{CLE}_4$ (Sheffield-Werner'2001). As a byproduct, we derive that the number of pivotal edges for the one-arm event (i.e., the sign cluster containing the origin has diameter at least $N$) is typically of order $N^{(\frac{d}{2}-1)\land 2}$. This immediately implies that for the incipient infinite cluster (IIC) of the metric graph GFF, the dimension of cut edges (i.e., edges whose removal disconnects the IIC) equals $(\frac{d}{2}-1)\land 2$. Translated in the language of critical loop soups (whose clusters, by the isomorphism theorem, have the same distribution as GFF sign clusters), this leads to the analogous estimates where the counterpart of a pivotal edge is a pivotal loop at scale $1$. This result hints at the new and possibly surprising idea that already in dimension $3$, microscopic loops (even those at scale $1$) play a crucial role in the construction of macroscopic loop clusters.