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

TL;DR

This paper investigates the asymptotic behavior of the critical threshold in stretched-out hyperbolic random connection models, using lace expansion techniques and analyzing spectral radii, with applications to Boolean disc and heat kernel RCMs.

Contribution

It introduces a model-dependent analysis of the critical threshold in hyperbolic RCMs, highlighting qualitative differences from Euclidean cases and applying the results to specific models.

Findings

Critical threshold behavior analyzed for hyperbolic RCMs

Lace expansion used to evaluate spectral radii

Applications to Boolean disc and heat kernel RCMs

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.