Betti numbers in the Random Connection Model for higher-dimensional simplicial complexes and the Boolean model

TL;DR

This paper introduces a generalized random simplicial complex model extending the Random Connection Model, derives a central limit theorem for Betti numbers, and applies it specifically to the Boolean model.

Contribution

It develops a unified framework for analyzing Betti numbers in higher-dimensional random complexes, including a CLT for these topological invariants.

Findings

Central limit theorem established for Betti numbers in the model

Unified approach encompasses several existing models as special cases

Application of CLT to the Boolean model's Betti numbers

Abstract

Random simplicial complexes, as generalizations of random graphs, have become increasingly popular in the literature in recent years. In this paper, we consider a new model for a random simplicial complex that was introduced in arXiv:2506.11918, which generalizes the Random Connection Model in a natural way and includes several models used in the literature as special cases. We focus on the marked stationary case with vertices in $R^d\times A$, where the mark space $A$ is an arbitrary Borel space. We will derive a central limit theorem for an abstract class of functionals and show that many of the typical functionals considered in the study of simplicial complexes, such as Betti numbers, fall into this class. As an important special case, we obtain a central limit theorem for Betti numbers in the Boolean model.