The Poisson stick model in hyperbolic space

In this paper we study the Poisson stick model in two dimensional hyperbolic space $\mathbb{H}^2,$ where the sticks all have length $L.$ Typically, percolation models in hyperbolic space undergo two phase transitions as the intensity $\lambda$ varies, namely the percolation phase transition and the uniqueness phase transition. For the Poisson stick model, the critical intensities at which these transitions occur will depend on $L$, and in this paper we study the asymptotic behavior of these critical points as $L\to \infty.$ Our main results show that the critical point for the percolation phase transition scales like $L^{-2},$ while the critical point for the uniqueness phase transition scales like $L^{-1}.$ Comparing these results to the analogous results in Euclidean space show that the behavior of the percolation phase transition is the same in these two settings, while the uniqueness phase transition scales differently.