Mean-field behaviour of the random connection model on hyperbolic space

TL;DR

This paper investigates the phase transition and critical behavior of the random connection model on hyperbolic space, showing that the critical exponents match mean-field predictions through geometric analysis.

Contribution

It identifies critical exponents for hyperbolic percolation and demonstrates their agreement with mean-field values using isoperimetric properties, a novel approach.

Findings

Existence of a critical intensity  for percolation in hyperbolic space.

Critical exponents match mean-field percolation values.

Critical clusters characterized via isoperimetric inequalities.

Abstract

We study the random connection model on hyperbolic space $\mathbb{H}^d$ in dimension $d=2,3$. Vertices of the spatial random graph are given as a Poisson point process with intensity $\lambda>0$. Upon variation of $\lambda$ there is a percolation phase transition: there exists a critical value $\lambda_c>0$ such that for $\lambda<\lambda_c$ all clusters are finite, but infinite clusters exist for $\lambda>\lambda_c$. We identify certain critical exponents that characterize the clusters at (and near) $\lambda_c$, and show that they agree with the mean-field values for percolation. We derive the exponents through isoperimetric properties of critical percolation clusters rather than via a calculation of the triangle diagram.