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

Abstract

As generalizations of random graphs, random simplicial complexes have attracted growing attention in the literature. In this paper, we introduce a new random simplicial complex that extends the Random Connection Model (RCM), a random graph model that has been extensively studied for over three decades, to higher-dimensional simplicial complexes. The resulting model allows simplices of different dimensions to be governed by separate connection functions, providing a higher-dimensional analogue of the RCM. For this model, we derive explicit moment formulas for a generalized Euler characteristic and establish quantitative central limit theorems under increasing intensity and increasing observation windows. To this end, we extend existing normal approximation results for Poisson functionals to a class of generalized difference operators. These results are obtained in a general framework in which the vertices of the simplicial complex are drawn from an arbitrary Borel space. In the stationary Euclidean setting with marks, we additionally establish a multivariate central limit theorem for simplex counts.