Heterochromatic two-arm probabilities for metric graph Gaussian free fields

TL;DR

This work analyzes heterochromatic two-arm connections for the Gaussian free field on the metric graph of $\mathbb{Z}^d$, establishing the sharp decay $\mathbb{P}(\mathsf{H}_{v}^{v'}(N)) \asymp N^{-[(\frac{d}{2}+1)\boxdot 4]}$ for $d\ge3$, $d\neq6$, with a distance-dependent prefactor in the small-distance regime. The authors leverage the loop-soup representation and Werner's switching identity, complemented by rigidity and separation lemmas, to obtain precise asymptotics for both two-arm and four-point heterochromatic events and to prove stability/regularity properties of two-cluster coexistence. They further show that conditioning on heterochromatic coexistence preserves the intrinsic volume-growth rates $M^{(\frac{d}{2}+1)\boxdot 4}$ for each cluster, establishing both lower and upper bounds for cluster volumes. The results illuminate how macroscopic sign clusters interact in high dimensions and contribute to the broader understanding of incipient infinite cluster phenomena on metric-graph GFFs.

Abstract

For the Gaussian free field on the metric graph of $\mathbb{Z}^d$ ($d\ge 3$), we consider the heterochromatic two-arm probability, i.e., the probability that two points $v$ and $v'$ are contained in distinct clusters of opposite signs with diameter at least $N$. For all $d\ge 3$ except the critical dimension $d_c=6$, we prove that this probability is asymptotically proportional to $N^{-[(\frac{d}{2}+1)\land 4]}$. Furthermore, we prove that conditioned on this two-arm event, the volume growth of each involved cluster is comparable to that of a typical (unconditioned) cluster; precisely, each cluster has a volume of order $M^{(\frac{d}{2}+1)\land 4}$ within a box of size $M$.

Figures (1)

• Figure 1: In this illustration, the cyan (resp. pink) regions represent the disjoint clusters $\mathcal{C}_{w_1}$ and $\mathcal{C}_{x_1}$ (resp. $\mathcal{C}_{w_2}$ and $\mathcal{C}_{x_2}$). In order to connect these two pairs of clusters separately, the loops $\widetilde{\ell}_1$ and $\widetilde{\ell}_2$ (i.e., the green and purple curves) are added to $\widetilde{\mathcal{L}}_{1/2}$. For $j\in \{1,2\}$, since $\widetilde{\ell}_j$ is contained in $D_j^{\epsilon_\ddagger\cdot \delta_{\ddagger}}$, the absence of the event $\mathsf{F}_{\epsilon_\ddagger}$ ensures that after adding $\widetilde{\ell}_j$, the enlarged part of the new cluster $\overline{\mathcal{C}}_j$ containing $x_j$ (i.e., the union of clusters intersecting $\widetilde{\ell}_j$) is contained in $D_j^{\delta_{\ddagger}}$ and hence, is disjoint from $\overline{\mathcal{C}}_{3-j}$. Thus, after this modification, the event $\mathsf{C}[(x_1,x_2,w_1,w_2)]$ occurs.

Theorems & Definitions (70)

• Theorem 1.1
• Remark 1.2: distance between macroscopic clusters
• Remark 1.3: pivotal edge
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Lemma 3.1
• proof
• Lemma 3.2
• proof