Thick points of log-correlated Gaussian fields do not depend on the mollifier

Abstract

The log-correlated Gaussian field (LGF) on $\mathbb{R}^d$ ($d \geq 2$) is a centered Gaussian random tempered distribution, defined modulo additive constants, whose covariance kernel is $\log(1/|x-y|)$. In two dimensions, the LGF coincides with the whole-plane Gaussian Free Field (GFF). Because the field is a distribution, studying its pointwise behavior requires regularization via convolution with a mollifier. A point is called $\alpha$-thick if the mollified field at that point grows like $\alpha \log(1/\epsilon)$ as the mollification scale $\epsilon \to 0$. A natural question is whether the set of $\alpha$-thick points depends on the choice of mollifier. We prove that for any two admissible mollifiers $\rho$ and $\sigma$ satisfying some mild conditions, the thick point sets coincide almost surely. The result holds in all dimensions $d \geq 2$, and in the special case $d = 2$ extends to the zero-boundary GFF on any open domain with harmonically non-trivial boundary via the Markov property.