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Thick points of log-correlated Gaussian fields do not depend on the mollifier — arXiv:2609.08132

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.08132
License (as archived): CC0-1.0
Archived: 2026-10-16T14:35:11.216Z
SHA-256 of the archived text: eba7574f8aed7dc1c65ab0c7c4790bd38f0759a633e96d46f667e2eeb1e4297a
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How to cite

Thick points of log-correlated Gaussian fields do not depend on the mollifier — arXiv:2609.08132 — arXiv (en), https://arxiv.org/abs/2609.08132 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_862b7dc87fbd (Metraipolis archive dc_862b7dc87fbd, sha256 eba7574f8aed7dc1c65ab0c7c4790bd38f0759a633e96d46f667e2eeb1e4297a)

BibTeX

@misc{dc862b7dc87fbd,
  title = {Thick points of log-correlated Gaussian fields do not depend on the mollifier — arXiv:2609.08132},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.08132},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_862b7dc87fbd; sha256 eba7574f8aed7dc1c65ab0c7c4790bd38f0759a633e96d46f667e2eeb1e4297a},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.08132. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-09 in math.PR).
Categorie arXiv: Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2609.08132 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T14:09:59.792Z dalla pagina https://arxiv.org/catchup/math.PR/2026-09-09?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).