标题： 关于布朗交错的可视窗口、泊松圆柱体和布尔模型 显示英文标题

标题： On the visibility window for Brownian interlacements, Poisson cylinders and Boolean models

摘要： 我们研究三种在$\mathbb R^d$中具有空间相关性缓慢衰减的模型内的可见性：布朗交错，泊松圆柱体和泊松-布尔模型。 令$Q_x$为以$x$为中心的最大的球的半径，该球的每个点都可以通过这些模型之一的空集从$0$可见。 我们证明了在 $x$ 从 $0$ 可见的条件下，当 $x\to\infty$ 时， $Q_x/\delta_{\|x\|}$ 弱收敛到具有显式强度的指数分布，该强度取决于各自模型的参数。 缩放函数 $\delta_r$ 是在 arXiv:2304.10298 中引入的可见窗口，是距离 $0$ 为 $r$ 的可见集中的相关性长度尺度。 显示英文摘要

摘要： We study visibility inside the vacant set of three models in $\mathbb R^d$ with slow decay of spatial correlations: Brownian interlacements, Poisson cylinders and Poisson-Boolean models. Let $Q_x$ be the radius of the largest ball centered at $x$ every point of which is visible from $0$ through the vacant set of one of these models. We prove that conditioned on $x$ being visible from $0$, $Q_x/\delta_{\|x\|}$ converges weakly, as $x\to\infty$, to the exponential distribution with an explicit intensity, which depends on the parameters of the respective model. The scaling function $\delta_r$ is the visibility window introduced in arXiv:2304.10298, a length scale of correlations in the visible set at distance $r$ from $0$.