Random Trees in Hyperbolic FPP

We start by surveying known properties of first passage percolation (FPP) geodesics on a Gromov-hyperbolic group $G$. Due to a coalescence phenomenon established in earlier work of the authors, a random tree $T(\xi ,\omega )$ consisting of infinite random geodesics to a point $\xi$ on the Gromov boundary $\mathrm{\partial }G$ emerges naturally. These trees have a rich random geometry that we explore further in this paper.