Random connection models in the thermodynamic regime: central limit theorems for add-one cost stabilizing functionals

Abstract

The paper deals with a random connection model, a random graph whose vertices are given by a homogeneous Poisson point process on ${\mathbb{R}}^d$, and edges are independently drawn with probability depending on the locations of the two end points. We establish central limit theorems (CLT) for general functionals on this graph under minimal assumptions that are a combination of the weak stabilization for the add-one cost and a $(2+\mathit{\delta })$-moment condition. As a consequence, CLTs for isomorphic subgraph counts, isomorphic component counts, the number of connected components are then derived. In addition, CLTs for Betti numbers and the size of the largest component are also proved for the first time.