Infinite geodesics in hyperbolic random triangulations

We study the structure of infinite geodesics in the Planar Stochastic Hyperbolic Triangulations ${\mathbb{T}}_{\lambda }$, which are the hyperbolic analogs of the UIPT. We prove that these geodesics form a supercritical Galton–Watson tree with geometric offspring distribution. The tree of infinite geodesics in ${\mathbb{T}}_{\lambda }$ provides a new notion of boundary, which is a realization of the Poisson boundary. By scaling limits arguments, we also obtain a description of the tree of infinite geodesics in the hyperbolic Brownian plane. Finally, by combining our main result with a forthcoming paper, we obtain new hyperbolicity properties of ${\mathbb{T}}_{\lambda }$: it satisfies a weaker form of Gromov-hyperbolicity and admits bi-infinite geodesics.