Arm exponent for the Gaussian free field on metric graphs in intermediate dimensions

TL;DR

The paper analyzes bond percolation on metric-graph graphs driven by Gaussian free field excursion sets in intermediate dimensions (1 ≤ ν ≤ α/2). It develops obstacle-set and loop-soup/interlacement methods to tightly bound critical and near-critical observables. The authors prove that the critical one-arm probability decays as R^{-ν/2+o(1)} with precise logarithmic corrections depending on ν, and obtain near-sharp upper bounds on truncated two-point functions, enabling a clear characterization of the correlation length via 2/ν. These results extend previous work to a broad class of graphs with polynomial volume growth and power-law Green’s function decay, and they elucidate rotational-invariance-like behavior in Z^3 at criticality. The methods unify loop-soup techniques, interlacement couplings, and capacity-based estimates to sharpen understanding of critical and near-critical percolation in non-Euclidean settings.

Abstract

We investigate the bond percolation model on transient weighted graphs ${G}$ induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in ${G}$ have polynomial volume growth with growth exponent $α$ and that the Green's function for the random on ${G}$ exhibits a power law decay with exponent $ν$, in the regime $1\leq ν\leq \fracα{2}$. In particular, this includes the cases of ${G}=\mathbb{Z}^{3}$ for which $ν=1$, and ${G}= \mathbb{Z}^{4}$ for which $ν=\fracα{2}=2$. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance $R$, like $R^{-\fracν{2}+o(1)}$. Our results are, in fact, more precise and yield logarithmic corrections when $ν>1$ as well as corrections of order $\log \log R$ when $ν=1$. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when $ν>1$ and essentially optimal when $ν=1$. This extends previous results from arXiv:2101.05801 and arXiv:1807.11117.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Lemma 3.1: $0< \nu \leq \frac{\alpha}{2}$
• proof