arXiv subjects

Dominik Pabst

Publications and source records attributed to Dominik Pabst.

4 recordsLinked to original sources

arXiv2025

Percolation in the marked stationary Random Connection Model for higher-dimensional simplicial complexes

We introduce a novel percolation model that generalizes the classical Random Connection Model (RCM) to a random simplicial complex, allowing for a more refined understanding of connectivity and emergence of large-scale structures in random topological spaces. Regarding percolation with respect to the notion of up-connectivity, we establish the existence of a sharp phase transition for the appearance of a giant component, akin to the well-known threshold behavior in random graphs. This sharp phase transition is, in its generality, new even for the classical RCM as a random graph. As special cases, we obtain sharp phase transitions for the Vietoris-Rips complex, the Cech complex, and the Boolean model, allowing us to identify which properties of these well-known percolation models are actually required.

math.PR↗

arXiv2025

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

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.

math.PR↗

arXiv2025

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

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.

math.PR↗

arXiv2025

Minkowski tensors for point clouds and voxelized data: robust, asymptotically unbiased estimators

Minkowski tensors, also known as tensor valuations, provide robust $n$-point information for a wide range of random spatial structures. Local estimators for point clouds, e.g., representing voxelized data, however, are unavoidably biased even in the limit of infinitely high resolution. Here, we substantially improve a recently proposed, asymptotically unbiased algorithm to estimate Minkowski tensors from point clouds. Our improved algorithm is more robust and efficient. Moreover we generalize the theoretical foundations for an asymptotically bias-free estimation of the interfacial tensors, among others, to the case of finite unions of compact sets with positive reach, which is relevant for many applications like rough surfaces or composite materials. As a realistic test case of random spatial structures, we consider random (beta) polytopes. We first derive explicit expressions of the expected Minkowski tensors, which we then compare to our simulation results. We obtain precise estimates with relative errors of a few percent for practically relevant resolutions. Finally, we apply our methods to real data of metallic grains and nanorough surfaces, and we provide an open-source python package, which works in any dimension.

math.ST↗