Stationary random subgroups in negative curvature

We show that discrete stationary random subgroups of isometry groups of Gromov hyperbolic spaces have full limit sets as well as critical exponents bounded from below. This information is used to answer a question of Gelander and show that a rank one locally symmetric space for which the bottom of the spectrum of the Laplace-Beltrami operator is the same as that of its universal cover has unbounded injectivity radius.