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

TL;DR

This paper introduces a new percolation model for random simplicial complexes, establishing sharp phase transitions for connectivity, extending classical results to higher-dimensional topological spaces and specific models like Vietoris-Rips and Cech complexes.

Contribution

It generalizes the Random Connection Model to simplicial complexes and proves the existence of sharp phase transitions for percolation in this broader setting.

Findings

Existence of a sharp phase transition for giant component emergence

Application to Vietoris-Rips and Cech complexes

Identification of properties needed for percolation in these models

Abstract

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.