First Passage Percolation on Hyperbolic groups

Abstract: We study first passage percolation (FPP) on a Gromov-hyperbolic group $G$ with boundary $\partial G$ equipped with the Patterson-Sullivan measure $\nu$. We associate an i.i.d.\ collection of random passage times to each edge of a Cayley graph of $G$, and investigate classical questions about the asymptotics of first passage time as well as the geometry of geodesics in the FPP metric. Under suitable conditions on the passage time distribution, we show that the `velocity' exists in $\nu$-almost every direction $\xi\in \partial G$, and is almost surely constant by ergodicity of the $G-$action on $\partial G$. For every $\xi\in \partial G$, we also show almost sure coalescence of any two geodesic rays directed towards $\xi$. Finally, we show that the variance of the first passage time grows linearly with word distance along word geodesic rays in every fixed boundary direction. This provides an affirmative answer to a conjecture of Benjamini, Tessera, and Zeitouni.