We consider the Poisson cylinder model in $d$-dimensional hyperbolic space. We show that in contrast to the Euclidean case, there is a phase transition in the connectivity of the collection of cylinders as the intensity parameter varies. We also show that for any non-trivial intensity, the diameter of the collection of cylinders is infinite.
"Poisson cylinders in hyperbolic space." Electron. J. Probab. 20 1 - 25, 2015. https://doi.org/10.1214/EJP.v20-3645