Boolean models

TL;DR

This survey reviews the geometric functionals of Boolean models in Euclidean space, focusing on their properties, formulas, and limit theorems, with applications in physics, materials science, and biology.

Contribution

It provides a comprehensive overview of geometric functionals of Boolean models, including density formulas, second order properties, and central limit theorems, highlighting recent advances.

Findings

Derived local and asymptotic density formulas.

Analyzed second order properties of Boolean models.

Established central limit theorems for geometric functionals.

Abstract

The topic of this survey are geometric functionals of a Boolean model (in Euclidean space) governed by a stationary Poisson process of convex grains. The Boolean model is a fundamental benchmark of stochastic geometry and continuum percolation. Moreover, it is often used to model amorphous connected structures in physics, materials science and biology. Deeper insight into the geometric and probabilistic properties of Boolean models and the dependence on the underlying Poisson process can be gained by considering various geometric functionals of Boolean models. Important examples are the intrinsic volumes and Minkowski tensors. We survey here local and asymptotic density (mean value) formulas as well as second order properties and central limit theorems.