Estimating the critical threshold for stretched-out hyperbolic random connection models

2025/11/15 by Matthew Dickson, Dickson, Matthew
Mathematics · Physics and Astronomy · #43A90 #60G55 #82B43 #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2511.11974

openalex publication_date 2025/11/15 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28

Abstract

This paper examines the model-dependent asymptotic behaviour of the critical threshold intensity for stretched-out random connection models (RCMs) on hyperbolic spaces. The proof uses lace expansion arguments, but has notable qualitative differences to the Euclidean case in how it evaluates spectral radii. The result is applied to the Boolean disc RCM and a heat kernel RCM.

Citations

• Non-Uniqueness Phase in Hyperbolic Marked Random Connection Models using the Spherical Transform
• Spread-out percolation on transitive graphs of polynomial growth
• On the uniqueness of the infinite cluster and the cluster density in the Poisson driven random connection model
• A lower bound for p_c in range-R bond percolation in four, five and six dimensions
• Critical Exponents for Marked Random Connection Models
• The Triangle Condition for the Marked Random Connection Model
• Expansion for the critical point of site percolation: the first three\n terms
• Lace Expansion and Mean-Field Behavior for the Random Connection Model
• Percolation on hyperbolic graphs
• A lower bound for p_c in range-R bond percolation in two and three dimensions
• Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions

Related

• Mean-field behaviour of the random connection model on hyperbolic space
• Lace Expansion and Mean-Field Behavior for the Random Connection Model
• Non-Uniqueness Phase in Hyperbolic Marked Random Connection Models using the Spherical Transform
• Critical Exponents for Marked Random Connection Models
• Supercritical phase of the random connection model