Critical level set percolation for the GFF in $d>6$: comparison principles and some consequences

TL;DR

The article analyzes critical level-set percolation for the Gaussian free field on $\mathbb{Z}^d$ in high dimensions ($d>6$) by leveraging a loop-soup representation and two geometric comparison schemes. It develops extrinsic-intrinsic metric comparisons and a non-backtracking/simple-path framework (via glued-loops and BLE/ITE) to obtain sharp intrinsic-geometry estimates, including a quadratic relation between chemical and Euclidean distances and the intrinsic one-arm exponent. Key contributions include a local connectivity bound, averaged loop-diameter removal results in the loop-percolation model, and a rigorous intrinsic-arm analysis that matches mean-field predictions, extending results previously known only in higher dimensions. The methods yield robust tools for long-range percolation models arising from GFF level sets, with potential implications for the Alexander-Orbach conjecture and related percolation phenomena. Overall, the work advances mean-field understanding of GFF-level-set percolation down to $d>6$ and introduces versatile geometric techniques for analyzing loop-based percolation.

Abstract

The intrinsic geometry of the critical percolation cluster induced by the level set of the metric Gaussian free field on $\mathbb{Z}^{d}$ has been the subject of much recent activity. (Lupu, 2016) established that the critical percolation cluster has the same law as that in a Poisson loop soup where the intensity is dictated by the Green's function of the usual random walk. A sharp Euclidean one arm exponent was proven recently in (Cai and Ding, 2023), and subsequently in (Ganguly and Nam, 2024) other results about the chemical one arm exponent, volume growth and the Alexander-Orbach conjecture were established for all $d>20.$ In this article, we introduce new methods to obtain several sharp estimates about the intrinsic geometry which hold for all $d>6$. We develop two primary comparison methods. The first involves a comparison of the extrinsic (Euclidean) and intrinsic metrics allowing estimates about the former to be transferrable to the latter. The second compares intrinsic geodesics to a class of modified paths which are non-backtracking in a certain loop sense. The latter is amenable to analysis using methods of bond percolation. We finally prove that such non-backtracking paths are not much longer than geodesics allowing us to establish comparable estimates for the geodesic. As applications of such methods, we establish, for all $d>6,$ the chemical one-arm exponent, as well as an averaged version of a conjecture from (Werner, 2021) which asserts that deletion of loops with diameter larger than $r^{6/d}$ do not affect the connection probabilities between points at Euclidean distance $r.$ An important ingredient of independent interest is a local connectivity estimate which asserts that the connection probability between two points at distance $r$ remains, up to constants, unchanged even if the connecting path is confined to a ball of diameter comparable to $r.$

Figures (3)

• Figure 1: Illustration of the above lemma. On the left is a path running through various loops, on the right, a subsequence of these loops is highlighted in bold, forming a simple chain.
• Figure 2: Illustration of an $x$-over-counted point $z$. $x$ and $z$ lie on two distinct simple paths connecting $0$ to $\partial \mathbb{B}_{r}$ with different end points on $\partial \mathbb{B}_{r}$. These two paths share a common segment (highlighted in orange) at the beginning until this orange segment enters $\Gamma_{2}$ from $x$. Beyond this, the path splits into two separate segments: a green one and a red one. The red path passes through $\Gamma$, where $z$ is located.
• Figure 3: On the left is an illustration of $A_{1}(u,v)$, where $u$ and $v$ belong to two distinct loops $\Gamma$ and $\Gamma^{\prime}$ in $\mathsf{Set}(\overline{L}_{1})$. The right figure illustrates $A_{2}(u,v)$, where $u$ and $v$ are located on the same loop $\Gamma$.

Theorems & Definitions (61)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1: Proposition 2.1 in lupu2016loop
• Lemma 2.2
• Lemma 2.3: BKR inequality
• Lemma 2.4
• Lemma 2.5: Tree expansion