Let G be a rank one simple Lie group or an hyperbolic group. In a joint work with Ilya Gekhtman we show that non-free stationary actions of G have "large" stabilizers - if the stabilizers are discrete then they have full limit sets and critical exponents bounded away from 0. As an application, we use random walk techniques to obtain a conditional analogue of the recent theorem of Fraczyk-Gelander in rank one. Namely, if the bottom of the spectrum of the Laplacian on the hyperbolic manifold M is equal to that of its universal cover then M has points with arbitrary large injectivity radius.