任意维高斯自由场的陡点
Steep Points of Gaussian Free Fields in Any Dimension
摘要：
本文将高斯自由场（GFF）的经典厚点集的豪斯多夫维数的已有结果推广到更一般的例外集.本文采用圆或球平均正则化方法处理任意维奇异GFF，并引入了GFF的“$f-$陡点”的概念，给出了检验函数$f$的定义.粗略地说，GFF的一般元素的$f-$陡峭点是当由函数$f$加权时正则化的场元素的“陡峭度”或换句话说“变化率”变得异常大的位置。不同的选择$f$导致GFF的各种异常行为的研究。我们研究了由$f-$陡点组成的集合的豪斯多夫维数，从这个维数中我们可以恢复现有的关于对数相关和多项式相关GFF的厚点集的结果，并且在例外集上也得到了新的结果，据我们所知，这些例外集以前没有被研究过.我们的方法的灵感来自于用于研究经典的2D对数相关GFF的厚点集。
英文摘要：
This work aims to extend the existing results on the Hausdorff dimension of the classical thick point sets of a Gaussian free field (GFF) to a more general class of exceptional sets. We adopt the circle or sphere averaging regularization to treat a singular GFF in any dimension, and introduce the notion of "$f-$steep point" of the GFF for certain test function $f$. Roughly speaking, the $f-$steep points of a generic element of the GFF are locations where, when weighted by the function $f$, the "steepness", or in other words, the "rate of change" of the regularized field element becomes unusually large. Different choices of $f$ lead to the study of various exceptional behaviors of the GFF. We investigate the Hausdorff dimension of the set consisting of $f-$steep points, from which we can recover the existing results on thick point sets for both log-correlated and polynomial-correlated GFFs, and also obtain new results on exceptional sets that, to our best knowledge, have not been previously studied. Our method is inspired by the one used to study the thick point sets of the classical 2D log-correlated GFF.
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