Chemical Distance for the Level Sets of the Gaussian Free Field

TL;DR

The paper analyzes the chemical distance in the supercritical level-set percolation of the Gaussian free field on $\\mathbb{Z}^d$ ($d\ge 3$). It develops a renormalization-bootstrap framework based on the Gibbs–Markov decomposition $\\varphi = \\psi^U + \\xi^U$, constructing good/bad boxes to control connectivity and deriving sharp upper and lower bounds for the probability that chemical distance exceeds the Euclidean scale. The main results establish an upper bound of $\\mathbb{P}[\exists x,y: \rho_h(x,y) > C N] \leq \exp(-c N^{1-2/d})$ for $h<h_*$, and matching-type lower bounds $\\mathbb{P}[\exists x,y: \rho_h(x,y) > \alpha N] \geq \\exp(-c N/\\log N)$ in $$d=3$ (and similar in $d\ge 4$), clarifying the exponent gap and the role of long-range dependence. The approach combines multiscale decoupling, capacity estimates, and path-amendment arguments to transfer local connectivity into global control, contributing to a deeper understanding of geometric properties of correlated percolation models.

Abstract

We consider the Gaussian free field $\varphi$ on $\mathbb{Z}^d$ for $d \geq 3$ and study the level sets $\{\varphi \geq h \}$ in the percolating regime. We prove upper and lower bounds for the probability that the chemical distance is much larger than Euclidean distance. Our proof uses a renormalization scheme combined with a bootstrap argument.

Figures (2)

• Figure 1: The red boxes represent bad boxes, while the blue line represents a connected subset of the level sets that connects the origin to $Ne_1$. Crossing a bad box, which is unavoidable in this example, incurs a cost of $O(L^d)$, whereas avoiding bad boxes costs $O(L)$.
• Figure 2: The event $D_N(h)$ creates a long path (in blue) inside $U_N \cap E^{\geq h}$, while the event $F_N(h)$ insulates (in red) the path, forcing the chemical distance to be at least $\alpha N$.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Lemma 2.2: SznitmanDisconnection
• Lemma 2.3: SznitmanDisconnection
• Definition 2.4
• Lemma 2.5: DrewitzRathSapozhnikovChemical,EqualityParametersGFF
• Lemma 2.6: DrewitzRathSapozhnikovChemical,EqualityParametersGFF
• Definition 3.1
• Definition 3.2