Coverage and connectivity in stochastic geometry

需要统计分析的几何模式出现在许多科学和技术分支中，空间概率建模在材料科学和电信等不同领域都很重要。随机几何是对这些几何概率模型的数学分析。布尔模型是随机几何中的一个基本模型，它的一个版本如下。考虑一组小的（可能重叠的）“水滴”集中在一个光滑有界的空间区域（欧几里得空间，或者更普遍的黎曼流形）中一个大的随机“点云”的点上。在适当的数学假设下，我们可以提出如下问题。由液滴并集给出的随机集，完全覆盖它们所在区域的概率是多少？这个随机集合连通的概率是多少？如果没有连接，它分成多少个组件？给定空间中两个不动的点，它们通过这个随机集合的路径连接起来的概率是多少？这个随机集合的补连通的概率是多少？回答这类问题的精确公式是没有的。此外，在应用程序中，样本量通常非常大。因此，我们建议在点云大而液滴小的适当限制制度下研究这些类型的问题。解决这些问题的大部分困难来自于必须处理点所在区域的边界。从本质上讲，这是因为靠近边界的任何位置都很难被覆盖，但这些位置的体积小于内部位置的体积，因此必须估计这两种影响之间的权衡。我们将开发新的方法来处理边界效应，至少在边界是光滑的情况下，从而对上述许多问题得出完整的答案；以前，这样的答案主要只能在没有边界的简单情况下得到（例如，在环面中）。我们的方法应与其他各种随机几何模型有关，也应与具有多面体边界的区域有关。平面区域中的随机点云可以表示无线发射机的集合。它们也与统计集估计和拓扑数据分析（TDA）有关，其中欧几里得空间或流形区域（可能是高维）中的随机点可能表示多元统计数据。在TDA中，人们的目标是从点云中了解底层空间的拓扑结构，通常是通过一个离散的结构，比如点上的图，在附近的点之间有边。

Geometrical patterns requiring statistical analysis arise in many branches of science and technology, and spatial probabilistic modelling is important in diverse areas such as materials science and telecommunications. Stochastic geometry is the mathematical analysis of these geometrical probabilistic models.A fundamental model in stochastic geometry is the Boolean model, a version of which goes as follows. Consider a collection of small (possibly overlapping) `droplets' centred on the points of a large random `point cloud' in a smoothly bounded region of space (either Euclidean space, or more generally a Riemannian manifold). Under appropriate mathematical assumptions, we may ask questions such as the following.What is the probability that the random set, given by the union of the droplets, fully covers the region into which they are placed? What is the probability that this random set is connected? If not connected, how many components does it split into? Given two fixed points in space, what is the probability that they are connected by a path through this random set? What is the probability that the complement of this random set is connected?Exact formulae to answer these kinds of question are not available. Moreover, in applications the sample size is often very large. Thus we propose to investigate these types of question, in appropriate limiting regimes where the point cloud is large and the droplets are small.Much of the difficulty in addressing these problems arises from having to handle the boundary of the region in which the points are placed. Essentially this is because it is harder for any location near the boundary to be covered, but the volume of such locations is less than the volume of interior locations so one has to estimate the trade-off between these two effects. We shall develop new methods to deal with boundary effects, at least when the boundary is smooth, thereby deriving complete answers to many of the questions above; previously, such answers have been mainly available only in simpler cases where there is no boundary (for example, in a torus). Our methods should be relevant to various other models of stochastic geometry, and also to regions with polyhedral boundariesThese kinds of problem are relevant to wireless communications; a random point cloud in a planar region may represent a collection of wireless transmitters. They are also relevant to statistical set estimation and to topological data analysis (TDA), where the random points in a region of (possibly high-dimensional) Euclidean space or manifold may represent multivariate statistical data. In TDA one aims to learn about the topology of the underlying space from the point cloud, often through a discrete structure such as a graph on the points with edges between nearby points.