  Let $X$ be a proper geodesic Gromov hyperbolic metric space and let $G$ be a
cocompact group of isometries of $X$ admitting a uniform lattice. Let $d$ be
the Hausdorff dimension of the Gromov boundary $\partial X$. We define the
critical exponent $\delta(\mu)$ of any discrete invariant random subgroup $\mu$
of the locally compact group $G$ and show that $\delta(\mu) > \frac{d}{2}$ in
general and that $\delta(\mu) = d$ if $\mu$ is of divergence type. Whenever $G$
is a rank-one simple Lie group with Kazhdan's property $(T)$ it follows that an
ergodic invariant random subgroup of divergence type is a lattice. One of our
main tools is a maximal ergodic theorem for actions of hyperbolic groups due to
Bowen and Nevo.
