发表机构
Captial Normal University; Sorbonne Université and Université Paris Cité, CNRS; Tsinghua University（首都师范大学; 索邦大学与巴黎 Cité 大学，法国国家科学研究中心; 清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究多边形域中带对数奇点的Gaussian自由场的水平线，证明其分布与无奇点情形绝对连续，导数由约化格林能量给出，并揭示奇点附近行为的显著差异。
AI 中文摘要
我们分析了多边形区域中带对数奇点的Gaussian自由场的水平线。我们关注（非零）奇点情形，并将其与无奇点情形进行比较。在奇点情形下，场的水平线的分布关于无奇点情形的分布是绝对连续的，其Radon-Nikodym导数与F. Viklund和D. W. Nyström~[VN24]引入的约化格林能量相关。这些水平线具有布朗运动表示。在无奇点情形中，该表示涉及布朗环和游弋的泊松点过程；而在奇点情形中，它额外包含布朗泡和桥的泊松点过程。一个或两个奇点的特殊情形产生了共形半径的矩生成函数，这可以解释为共形场论中自由玻色子点处顶点算子的一点和两点相关函数。虽然Radon-Nikodym导数在奇点附近有界，但水平线在奇点附近的行为与无奇点情形显著不同，所有水平线保持接近奇点的渐近概率也不同。
英文摘要
We analyze the level lines of the Gaussian Free Field with logarithmic singularities in polygonal domains. We focus on the (none-zero) singular case and compare it to the singular-free setting. The law of the level lines for the field in singular case is absolutely continuous with respect to the law for the singular-free case, with a Radon-Nikodym derivative related to the reduced Green's energy which is introduced by F. Viklund and D. W. Nyström~[VN24]. These level lines admit a Brownian motion representation. In the singular-free case, such representation involves Poisson point processes of Brownian loops and excursions; while for the singular case, it additionally includes Poisson point processes of Brownian bubbles and bridges. The special cases of one or two singularities yield moment-generating functions of the conformal radius, which can be interpreted as one-point and two-point correlation functions of vertex operators in conformal field theory at the free-boson point. While the Radon-Nikodym derivative is bounded away from the singularities, the behavior of the level lines near the singularities is markedly different from the singular-free case, with different asymptotic probabilities for all level lines to remain close to the singularity.
Comments30 pages, 2 figures
