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

TL;DR

The paper establishes that, in dimensions $d\ge 3$ with $d\neq 6$, macroscopic sign clusters of the Gaussian free field on the metric graph $\tilde{\bb Z}^d$ can come within microscopic graph distance inside a box of side $N$, with probability governed by critical exponents involving $d$. Central to the analysis is the isomorphism between the GFF and the loop-soup at intensity $1/2$, which equates GFF sign clusters with loop-clusters and enables control via Brownian excursions and four-arm events. The authors derive precise two-arm, four-arm, and pivotal-event bounds, and use first/second moment methods and switching identities to show that the minimal distance between macroscopic clusters is typically microscopic; they also determine the typical number of pivotal loops/edges at scale 1 and connect these to the dimension of cut-edges in the IIC. Together, these results reveal that microscopic loops play a crucial role in forming macroscopic loop clusters in dimension $d=3$, contrasting sharply with the 2D conformal loop framework. The findings illuminate the delicate balance between macroscopic connectivity and microscopic loop structure in high-dimensional GFF and loop-soup models, with implications for IIC geometry and critical loop soups.

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 leads to disconnection of 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.

Figures (3)

• Figure 1: In this illustration, the cyan region is the cluster $\mathcal{C}_{w_1}$, and the pink regions represent the clusters $\{\mathcal{C}_{z_j}\}_{j\in \{1,2,3,5\}}$. Note that on $\overline{\mathsf{A}}^{\mathcal{C}_{w_-}}_+(\bar{z};r,r_{k} )$, the latter four clusters are restricted in different areas, and thus are constructed by disjoint collection of loops. This implies that given $\mathcal{C}_{w_1}$, the four sub-events involved in $\overline{\mathsf{A}}^{\mathcal{C}_{w_-}}_+(\bar{z};r,r_{k} )$ are conditionally independent. In the next step (\ref{['newadd49']}), we fix $\{\mathcal{C}_{z_j}\}_{j\in \{1,2,3,5\}}$ and then decompose the cluster $\mathcal{C}_{w_1}$ via Lemma \ref{['lemma41']}.
• Figure 2: In this illustration, each colored region represents a loop cluster. The clusters $\mathcal{C}_{v_e^+}$ and $\mathcal{C}_{v_e^-}$ certify the four-point event $\{v_e^+ \xleftrightarrow{} z_1 \Vert v_e^-\xleftrightarrow{} z_1'\}$, whose probability has been estimated in the companion paper inpreparation_twoarm (see (\ref{['four_point_use']})). The same applies to $\mathcal{C}_{v_{e'}^+}$ and $\mathcal{C}_{v_{e'}^-}$. To ensure that the remaining clusters certify four-point events restricted to disjoint annuli, we use Corollary \ref{['coro28']} to decompose the clusters $\mathcal{C}_{w_-}$ and $\mathcal{C}_{z_3}$ around the boundaries $\partial B(r_k)$ and $\partial B(r_{k'})$ respectively (see (\ref{['413']}) and (\ref{['414']})). In the special case when $k=k'$, this decomposition is not needed since the clusters $\mathcal{C}_{z_3}$, $\mathcal{C}_{z_3'}$, $\mathcal{C}_{w_+}$ and $\mathcal{C}_{w_-}$ already meet the requirements. This necessitates the distinct definitions of $\widehat{\mathsf{A}}_{\mathrm{III}}^{D }$ for the cases $k=k'$ and $k>k'$.
• Figure 3: In this illustration, each colored region is a loop cluster. These clusters are arranged in disjoint balls and annuli in pairs such that each pair of clusters certifies a four-point event in $\{\mathsf{C}_j\}_{1\le j\le 5}$. In particular, when $k=k_\star$, as verified in the companion paper inpreparation_twoarm, imposing $\mathcal{C}_{w_-}$ (resp. $\mathcal{C}_{w_+}$) as an absorbing boundary typically does not alter the order of the probability of $z_5 \xleftrightarrow{} w_{+}$ (resp. $z_6'\xleftrightarrow{} w_{-}$), i.e., $R^{2-d}$. In light of this, we substitute the four-point event $\{z_5\xleftrightarrow{} w_{+} \Vert z_6'\xleftrightarrow{} w_{-}\}$ with the event $\{\mathbf{A}_\star^{z_5}=\mathbf{A}_\star^{z_6'}= 1\}$ to simplify the analysis, while changing the probability by only a constant factor.

Theorems & Definitions (58)

• Theorem 1.1
• Remark 1.2
• Remark 1.3: Bernoulli percolation
• Remark 1.4: construction of the scaling limit
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof