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

2025/06/18 by Dominik Pabst, Pabst, Dominik · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #05C80 (Secondary) #60D05 (Primary) 60G55 #60K35 #Complex Network Analysis Techniques #FOS: Mathematics #Probability (math.PR) #Topological and Geometric Data Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2506.15038

openalex publication_date 2025/06/18 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

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.

Citations

• Central limit theorems for the Euler characteristic in the Random Connection Model for higher-dimensional simplicial complexes
• On the uniqueness of the infinite cluster and the cluster density in the Poisson driven random connection model
• Critical Exponents for Marked Random Connection Models
• Face and cycle percolation
• The Triangle Condition for the Marked Random Connection Model
• Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees
• Random connection models in the thermodynamic regime: central limit theorems for add-one cost stabilizing functionals
• The random connection model and functions of edge-marked Poisson processes: second order properties and normal approximation
• Subcritical phase of d-dimensional Poisson-Boolean percolation and its vacant set
• Thresholds for vanishing of `Isolated' faces in random Čech and Vietoris-Rips complexes
• Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on ℝ^d
• Every decision tree has an influential variable

Cited by

• On strong sharp phase transition in the random connection model

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• Percolation on random graphs
• Merging percolation and classical random graphs: Phase transition in dimension 1
• Merging percolation on Z^d and classical random graphs: Phase transition
• Phase Transitions in Partially Structured Random Graphs
• Central limit theorems for the Euler characteristic in the Random Connection Model for higher-dimensional simplicial complexes