Fractal dimension of critical Gaussian free field sign clusters
Abstract
We study metric graph sign clusters of the Gaussian free field on a large class of transient weighted graphs G that are sufficiently low-dimensional, i.e. below the mean-field regime, and prove a macroscopic equivalence property: several natural ways to define macroscopic clusters - involving any of diameter, volume, or capacity functionals - are in fact the same. As a corollary, macroscopic clusters are characterized in low dimensions as those containing large loops in the corresponding loop soup picture, in contrast with the typical behavior in high dimensions. We then apply these results to the small-mesh scaling limit of critical clusters on εℤd, d=3,4,5, as ε ↓ 0. We prove that every subsequential scaling limit consists of sets with positive Brownian capacity whose Hausdorff and Minkowski dimensions are both equal to 1+d/2, as conjectured by Werner. We also derive two-point crossing estimates in the limit, and show that each limiting loop soup cluster is the closure of the union of the scaling limits of the macroscopic loops it contains. Remarkably, unlike with planar Bernoulli percolation for instance, the determination of the Hausdorff dimension does not emerge from direct calculations involving a limiting object (such as SLE6), but rather from the distinctive features of the discrete model itself.
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BibTeX
@misc{drewitz2026fractal,
title = {{Fractal dimension of critical Gaussian free field sign clusters}},
author = {Alexander Drewitz and Alexis Prévost and Pierre-François Rodriguez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03703},
eprint = {2610.03703},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03703},
}
RIS
TY - UNPB
TI - Fractal dimension of critical Gaussian free field sign clusters
AU - Alexander Drewitz
AU - Alexis Prévost
AU - Pierre-François Rodriguez
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03703
UR - https://arxiv.org/abs/2610.03703
AB - We study metric graph sign clusters of the Gaussian free field on a large class of transient weighted graphs G that are sufficiently low-dimensional, i.e. below the mean-field regime, and prove a macroscopic equivalence property: several natural ways to define macroscopic clusters - involving any of diameter, volume, or capacity functionals - are in fact the same. As a corollary, macroscopic clusters are characterized in low dimensions as those containing large loops in the corresponding loop soup picture, in contrast with the typical behavior in high dimensions. We then apply these results to the small-mesh scaling limit of critical clusters on εℤ^d, d=3,4,5, as ε ↓ 0. We prove that every subsequential scaling limit consists of sets with positive Brownian capacity whose Hausdorff and Minkowski dimensions are both equal to 1+d/2, as conjectured by Werner. We also derive two-point crossing estimates in the limit, and show that each limiting loop soup cluster is the closure of the union of the scaling limits of the macroscopic loops it contains. Remarkably, unlike with planar Bernoulli percolation for instance, the determination of the Hausdorff dimension does not emerge from direct calculations involving a limiting object (such as SLE_6), but rather from the distinctive features of the discrete model itself.
ER -