Central limit theorems for the Euler characteristic in the Random Connection Model for higher-dimensional simplicial complexes

TL;DR

This paper extends the Random Connection Model to higher-dimensional simplicial complexes, deriving central limit theorems for the Euler characteristic in a very general setting, broadening understanding of random topological structures.

Contribution

It introduces a generalized RCM for simplicial complexes and establishes quantitative CLTs for the Euler characteristic across various asymptotic regimes.

Findings

Derived CLTs for the Euler characteristic in the new model

Unified framework encompassing existing models as special cases

Applicable to vertices from arbitrary Borel spaces

Abstract

As generalizations of random graphs, random simplicial complexes have been receiving growing attention in the literature. In this paper, we naturally extend the Random Connection Model (RCM), a random graph that has been extensively studied for over three decades, to a random simplicial complex, recovering many models currently found in the literature as special cases. In this new model, we derive quantitative central limit theorems for a generalized Euler characteristic in various asymptotic scenarios. We will accomplish this within a very general framework, where the vertices of the simplicial complex are drawn from an arbitrary Borel space.