We introduce the first passage set (FPS) of constant level $$-a$$ - a of the two-dimensional continuum Gaussian free field (GFF) on finitely connected domains. Informally, it is the set of points in the domain that can be connected to the boundary by a path on which the GFF does not go below $$-a$$ - a . It is, thus, the two-dimensional analogue of the first hitting time of $$-a$$ - a by a one-dimensional Brownian motion. We provide an axiomatic characterization of the FPS, a continuum construction using level lines, and study its properties: it is a fractal set of zero Lebesgue measure and Minkowski dimension 2 that is coupled with the GFF $$\Phi $$ Φ as a local set A so that $$\Phi +a$$ Φ + a restricted to A is a positive measure. One of the highlights of this paper is identifying this measure as a Minkowski content measure in the non-integer gauge $$r \mapsto \vert \log (r)\vert ^{1/2}r^{2}$$ r ↦ | log ( r ) | 1 / 2 r 2 , by using Gaussian multiplicative chaos theory.

中文翻译：

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二维连续高斯自由场的第一通道集

我们介绍了有限连接域上二维连续高斯自由场 (GFF) 的恒定水平 $$-a$$ - a 的第一个通道集 (FPS)。非正式地，它是域中的一组点，可以通过 GFF 不低于 $$-a$$ - a 的路径连接到边界。因此，它是 $$-a$$-a 的第一次击球时间的二维模拟，通过一维布朗运动。我们提供了 FPS 的公理表征，使用水平线的连续结构，并研究了它的属性：它是零勒贝格测度和 Minkowski 维 2 的分形集，与作为局部的 GFF $$\Phi $$ Φ 耦合设置 A 使得 $$\Phi +a$$ Φ + a 限制于 A 是一个正测度。本文的亮点之一是将此测度确定为非整数规中的 Minkowski 内容测度 $$r \mapsto \vert \log (r)\vert ^{1/2}r^{2}$$ r ↦ | 日志 ( r ) | 1 / 2 r 2 ，通过使用高斯乘法混沌理论。