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

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.