对数相关高斯场的厚点不依赖于磨光函数
Thick points of log-correlated Gaussian fields do not depend on the mollifier
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中文总结 AI 辅助
本文证明对数相关高斯场的厚点集合不依赖于磨光函数的选择，适用于所有维度 $d \geq 2$，并推广到二维零边界高斯自由场。
中文摘要 AI 辅助
在 $\mathbb{R}^d$（$d \geq 2$）上的对数相关高斯场（LGF）是一个中心化的高斯随机缓增分布，定义在模加性常数下，其协方差核为 $\log(1/|x-y|)$。在二维情形下，LGF 与全平面高斯自由场（GFF）一致。由于该场是一个分布，研究其逐点行为需要通过卷积磨光函数进行正则化。若在磨光尺度 $\epsilon \to 0$ 时，某点处的磨光场增长为 $\alpha \log(1/\epsilon)$，则该点称为 $\alpha$-厚点。一个自然的问题是：$\alpha$-厚点的集合是否依赖于磨光函数的选择。我们证明，对于任意两个满足某些温和条件的可容许磨光函数 $\rho$ 和 $\sigma$，厚点集合几乎必然重合。该结果在所有维度 $d \geq 2$ 中成立，并且在特殊情形 $d = 2$ 下，通过马尔可夫性质推广到任意具有调和非平凡边界的开区域上的零边界 GFF。
英文摘要
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 $α$-thick if the mollified field at that point grows like $α\log(1/ε)$ as the mollification scale $ε\to 0$. 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 $d \geq 2$, 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.
