Expansion of the Critical Intensity for the Random Connection Model

TL;DR

This paper derives an asymptotic expansion for the critical percolation density in the random connection model as the space dimension grows, providing explicit first terms for various specific models.

Contribution

It introduces a rigorous asymptotic expansion for the critical density in high-dimensional random connection models, including explicit calculations for several kernels.

Findings

Explicit first expansion terms for Gilbert disk and hyper-cubic models

Asymptotic behavior of critical density as dimension increases

Rigorous derivation for Gaussian and Cauchy kernels

Abstract

We derive an asymptotic expansion for the critical percolation density of the random connection model as the dimension of the encapsulating space tends to infinity. We calculate rigorously the first expansion terms for the Gilbert disk model, the hyper-cubic model, the Gaussian connection kernel, and a coordinate-wise Cauchy kernel.