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

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 $\hat{I}»>0$. Upon variation of $\hat{I}»$ there is a percolation phase transition: there exists a critical value $\hat{I}{»}_c>0$ such that for $\hat{I}»<\hat{I}{»}_c$ all clusters are finite, but infinite clusters exist for $\hat{I}»>\hat{I}{»}_c$. We identify certain critical exponents that characterize the clusters at (and near) $\hat{I}{»}_c$, 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.

39 pages, 7 figures