论文标题
二维游览云和离散高斯自由场的渗透
Percolation for two-dimensional excursion clouds and the discrete Gaussian free field
论文作者
论文摘要
我们研究了偏移过程的渗透性特性和平面单元磁盘中离散的高斯自由场（DGFF）。我们将使用随机步道定义的离散游览云视为随机讲述的二维版本及其使用布朗运动定义的缩放限制。我们证明，这两个模型与空置集渗透相关的关键参数是相同的，等于$π/3。$该值是从Schramm-Loewner Evolution（SLE）计算中获得的。通过同构定理，我们使用离散结果的概括，还涉及循环汤（和SLE计算），以表明与DGFF的级别设置相关的关键参数严格为正面且小于$ \ sqrt {π/2}。 DGFF的临界渗透参数和二维偏移云。类似的严格不平等现象可以在一般的瞬态设置中持有。
We study percolative properties of excursion processes and the discrete Gaussian free field (dGFF) in the planar unit disk. We consider discrete excursion clouds, defined using random walks as a two-dimensional version of random interlacements, as well as its scaling limit, defined using Brownian motion. We prove that the critical parameters associated to vacant set percolation for the two models are the same and equal to $π/3.$ The value is obtained from a Schramm-Loewner evolution (SLE) computation. Via an isomorphism theorem, we use a generalization of the discrete result that also involves a loop soup (and an SLE computation) to show that the critical parameter associated to level set percolation for the dGFF is strictly positive and smaller than $\sqrt{π/2}.$ In particular this entails a strict inequality of the type $h_*<\sqrt{2u_*}$ between the critical percolation parameters of the dGFF and the two-dimensional excursion cloud. Similar strict inequalities are conjectured to hold in a general transient setup.
