First-passage percolation on hyperbolic groups: Geodesics, coalescence, and exceptional directions

I shall discuss first-passage percolation on Cayley graphs of Gromov hyperbolic groups under mild conditions on the passage time distribution. Appealing to deep geometric and topological facts about hyperbolic groups and their boundaries, several questions become more tractable in this set-up compared to their counterparts in Euclidean lattices. In particular, I shall describe several results about time constants, fluctuations, coalescence of geodesics, and exceptional directions where coalescence fails. Some of the results are parallel to what are expected in Euclidean background geometry, while substantially different features are exhibited in other aspects.

This is based on joint works with Mahan Mj.