标题： 二维遍历云的渗透和离散高斯自由场 显示英文标题

标题： Percolation for two-dimensional excursion clouds and the discrete Gaussian free field

摘要： 我们研究平面单位圆盘中渗流过程和离散高斯自由场（dGFF）的渗流性质。 我们考虑使用随机游走定义的离散穿越云，作为随机穿插的二维版本，以及使用布朗运动定义的其尺度极限。 我们证明了这两种模型的空集渗流相关临界参数相同，且等于$\pi/3.$该值通过施拉姆-洛瓦斯演化（SLE）计算得到。 通过一个同构定理，我们利用一个涉及环形汤（以及SLE计算）的离散结果的推广，证明了dGFF的水平集渗流相关临界参数严格为正且小于$\sqrt{\pi/2}.$特别地，这意味着dGFF与二维穿越云的临界渗流参数之间存在严格不等式$h_*<\sqrt{2u_*}$。 在一般的瞬态设置中，类似的严格不等式被猜想成立。 显示英文摘要

摘要： 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 $\pi/3.$ 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 $\sqrt{\pi/2}.$ 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.