First passage percolation (FPP) gives a well-known model of random geometry on a fixed background infinite graph. When we specialize to Cayley graphs of Gromov-hyperbolic groups $G$, random trees $T$ emerge naturally. The first part of the talk will dwell on setting up hyperbolic FPP and outlining its basic properties. This will have a probabilistic emphasis.

In the second part, we will specialize to the study of exceptional directions, i.e. distinct random geodesics in $T$ that converge asymptotically to the same point in the boundary $\partial G$ of $G$. This will have a geometric group theoretic emphasis (joint work with Riddhipratim Basu).

If time permits, we will describe how to reconstruct the bulk random hyperbolic geometry from the boundary. Random trees leave a trace on the boundary $\partial G$ of $G$ in the form of an evolving random partition. It turns out that there is an inverse construction, where the random metric can be reconstructed (up to bounded errors) from the data of an evolving random partition on the boundary.