Quasi-multiplicativity and regularity for metric graph Gaussian free fields

TL;DR

The paper establishes quasi-multiplicativity for critical metric-graph GFF level-sets, showing that for $3\le d\le 5$ the probability of connecting two disjoint sets factors into the product of their individual connectivities, while for $d\ge 7$ a dimension-dependent correction factor $N^{6-d}$ appears (with $d=6$ exhibiting a polylogarithmic gap). This is achieved through a dual approach via the loop-soup isomorphism and detailed analysis of point-to-set and boundary-to-set connecting probabilities, including harmonic-averaged quantities, Harnack-type inequalities, and loop-cluster decompositions with BKR arguments. The results yield not only quasi-multiplicativity but also regularity properties of connecting probabilities and a comprehensive understanding of volume tails of critical clusters, with sharp bounds in $3\le d\le 5$ and conjectural precision in $d=6$. The insights align with conjectures for Bernoulli percolation and underpin the existence and equivalence of incipient infinite clusters (IIC) for metric-graph GFFs across dimensions, contributing to a broader percolation- Gaussian free field correspondence. The work thus provides rigorous tools for multi-scale connectivity analysis and advances the probabilistic theory of critical GFF percolation on metric graphs.

Abstract

We prove quasi-multiplicativity for critical level-sets of Gaussian free fields (GFF) on the metric graphs $\widetilde{\mathbb{Z}}^d$ ($d\ge 3$). Specifically, we study the probability of connecting two general sets located on opposite sides of an annulus with inner and outer radii both of order $N$, where additional constraints are imposed on the distance of each set to the annulus. We show that for all $d \ge 3$ except the critical dimension $d=6$, this probability is of the same order as $N^{(6-d)\land 0}$ (serving as a correction factor) times the product of the two probabilities of connecting each set to the closer boundary of this annulus. The analogue for $d=6$ is also derived, although the upper and lower bounds differ by a divergent factor of $N^{o(1)}$. Notably, it was conjectured by Basu and Sapozhnikov (2017) that quasi-multiplicativity without correction factor holds for Bernoulli percolation on $\mathbb{Z}^d$ when $3\le d<6$ and fails when $d>6$. In high dimensions (i.e., $d>6$), taking into account the similarity between the metric graph GFF and Bernoulli percolation (which was proposed by Werner (2016) and later partly confirmed by the authors (2024)), our result provides support to the conjecture that Bernoulli percolation exhibits quasi-multiplicativity with correction factor $N^{6-d}$. During our proof of quasi-multiplicativity, numerous regularity properties, which are interesting in their own right, were also established. A crucial application of quasi-multiplicativity is proving the existence of the incipient infinite cluster (IIC), which has been completed by Basu and Sapozhnikov (2017) for Bernoulli percolation for $3\le d<6$. Inspired by their work, in a companion paper, we also utilize quasi-multiplicativity to establish the IIC for metric graph GFFs, for all $d\ge 3$ except for the critical dimension $6$.

Figures (3)

• Figure 1: We consider the case $k=3$. The grey regions represent the target sets $\{A_j\}_{0\le j\le 3}$. The red (resp. blue) paths represent the forward (resp. backward) crossing paths from $\partial B(M)$ to $\partial B(N)$. The three black clusters are the explored clusters. As shown in this illustration, the event $\mathsf{F}_0$ in (\ref{['lastrevision5.9']}) is certified by the explored cluster $\hat{\mathcal{C}}_0$. In this example, the events $\mathsf{F}(\{1,2\})$ and $\mathsf{F}(\{3\})$ defined in (\ref{['lastrevision5.10']}) both happen and are certified by $\hat{\mathcal{C}}_1$ and $\hat{\mathcal{C}}_3$ respectively.
• Figure 2: In this illustration, the grey region represents the target set $A$. The black loop represents the loop $\widetilde{\ell}$ involved in the event $\mathsf{F}_{v_1,v_2,v_3}^{z_1,z_2}$ (recalling (\ref{['newdef_4.38']})). The blue, red and green clusters are three loop clusters that consist of disjoint collections of loops and certify $\{ v_1\xleftrightarrow{(D)}z_1 \}$, $\{ v_2\xleftrightarrow{(D)}z_2 \}$ and $\{ v_3\xleftrightarrow{(D)}A \}$ respectively. In this example, $v_1$ is contained in $B(M^{\frac{2}{d-4}})$ and hence, $\mathsf{F}^{z_1,z_2}_{(1)}$ happens, which further implies that $\{ z_1\xleftrightarrow{(D)}B(M^{\frac{2}{d-4}})\}$ (certified by the blue cluster) and $\{z_2\xleftrightarrow{(D)} A \}$ (certified by the union of the black loop and the red and green clusters) happen disjointly. In addition, the event $\mathsf{F}^{z_1,z_2}_{(3)}$ also occurs in this example because $v_3\in [\widetilde{B}(C_*N)]^c$.
• Figure 3: This is an illustration for the event $\mathsf{F}^{z_1,z_2}_{(4)}$. Since $v_1,v_2\in [\widetilde{B}(M^{\frac{2}{d-4}})]^c$ and $v_3\in \widetilde{B}(C_*N)$, the loop $\widetilde{\ell}$ involved in $\mathsf{F}_{v_1,v_2,v_3}^{z_1,z_2}$ must include forward and backward crossing paths (i.e., the red and blue paths) from $\partial B(\frac{1}{10}M^{\frac{2}{d-4}})$ to $\partial B(C_{\ref{['const_prepare_loop_decompose']}} C_*N)$. The three green clusters (with different shades) consist of disjoint collections of loops and certify $\{v_1\xleftrightarrow{(D)}z_1 \}$, $\{ v_2\xleftrightarrow{(D)}z_2 \}$ and $\{v_3\xleftrightarrow{(D)}A \}$ respectively.

Theorems & Definitions (66)

• Conjecture : basu2017kesten
• Theorem 1.1
• Remark 1.2: application of quasi-multiplicativity
• Remark 1.3: correction factor at the critical dimension $6$
• Remark 1.4: sharpness of the conditions for high dimensions
• Proposition 1.5
• Proposition 1.6
• Remark 1.7: extra conditions at the critical dimension $6$
• Proposition 1.8
• Proposition 1.9