论文标题
倾斜度和拓扑条件的标量高斯自由场之间的等效性
An equivalence between gauge-twisted and topologically conditioned scalar Gaussian free fields
论文作者
论文摘要
我们在公制图上研究了两种类型的标量高斯自由场（GFF），通常是$ \ { - 1,1 \} $ - 值的估值量规场。我们表明，可以通过对拓扑事件进行第一个调节，可以获得后者，直到额外的确定性转换。该事件是，该场的所有标志群都应该对量规场很重要，也就是说，不应包含具有固体$ -1 $的循环。我们还表达了这一拓扑事件的概率，这是拉普拉辛人的两个决定因素与$ 1/2 $的决定因素，通常是laplacian和the仪的laplacian。例如，这给出了环形平面域的概率，即公制图GFF的符号簇围绕着域的内孔。 基于我们在公制图上的结果以及Werner和Cai-ding的先前作品，并在高尺寸的指标GFF的群集上，我们制定了强度加倍的猜想。根据它，如果空间维度足够高，则指标图GFF的符号簇中的循环将缩放限制收敛于Brownian Loop loop参数$α= 1 = 1 = 1 = 1 = 2 \ times \ dfrac {1} {1} {2} $，这是强度参数的双重强度参数在异型中出现在异型中。
We study on the metric graphs two types of scalar Gaussian free fields (GFF), the usual one and the one twisted by a $\{-1,1\}$-valued gauge field. We show that the latter can be obtained, up to an additional deterministic transformation, by conditioning the first on a topological event. This event is that all the sign clusters of the field should be trivial for the gauge field, that is to say should not contain loops with holonomy $-1$. We also express the probability of this topological event as a ratio of two determinants of Laplacians to the power $1/2$, the usual Laplacian and the gauge-twisted Laplacian. As an example, this gives on annular planar domains the probability that no sign cluster of the metric graph GFF surrounds the inner hole of the domain. Based on our result on the metric graph, and on previous works by Werner and Cai-Ding on the clusters of the metric graph GFF in high dimension, we formulate an intensity doubling conjecture. According to it, if the space dimension is high enough, the cycles in the sign clusters of the metric graph GFF converge in the scaling limit to a Brownian loop soup of intensity parameter $α= 1 = 2\times \dfrac{1}{2}$, which is the double of the intensity parameter appearing in isomorphism theorems.
