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 boundaries
These 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.