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.
Categories: math.PR