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

TL;DR

This paper develops new comparison methods to analyze the intrinsic geometry of critical level set percolation clusters of the Gaussian free field in dimensions greater than six, leading to sharp estimates and confirming conjectures about connectivity and loop effects.

Contribution

It introduces two primary comparison techniques for intrinsic and extrinsic metrics and geodesics, applicable for all dimensions greater than six, advancing understanding of GFF percolation geometry.

Findings

Established the chemical one-arm exponent for all d>6.

Proved that removing large loops does not affect connection probabilities at scale r.

Provided a local connectivity estimate showing robustness of connection probabilities.

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 $r.$