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PreprintarXiv licencearXiv

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

  • Kyle Ambrose
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Abstract

The log-correlated Gaussian field (LGF) on ℝd (d ≥ 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 α-thick if the mollified field at that point grows like αlog(1/ε) as the mollification scale ε→ 0. A natural question is whether the set of α-thick points depends on the choice of mollifier. We prove that for any two admissible mollifiers ρ and σ satisfying some mild conditions, the thick point sets coincide almost surely. The result holds in all dimensions d ≥ 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.

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Cite

BibTeXRIS
BibTeX
@misc{ambrose2026thick,
  title = {{Thick points of log-correlated Gaussian fields do not depend on the mollifier}},
  author = {Kyle Ambrose},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08132},
  eprint = {2609.08132},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08132},
}
RIS
TY  - UNPB
TI  - Thick points of log-correlated Gaussian fields do not depend on the mollifier
AU  - Kyle Ambrose
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08132
UR  - https://arxiv.org/abs/2609.08132
AB  - The log-correlated Gaussian field (LGF) on ℝ^d (d ≥ 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 α-thick if the mollified field at that point grows like αlog(1/ε) as the mollification scale ε→ 0. A natural question is whether the set of α-thick points depends on the choice of mollifier. We prove that for any two admissible mollifiers ρ and σ satisfying some mild conditions, the thick point sets coincide almost surely. The result holds in all dimensions d ≥ 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.
ER  -