Large deviation principles for non-elementary random walks on hyperbolic spaces

We show that the displacement and translation distance of non-elementary random walks on isometry groups of hyperbolic spaces satisfy large deviation principles with the same rate function $I$. Roughly, this means that there exists function $I(t)$ which accurately predicts the exponential decay rate of the probability that the translation distance of a random product of length $n$ is $tn$, and similarly for the displacement. This settles a special case of a conjecture concerning the large deviation principle for the spectral radius of random matrix products. In a second part, we give a characterization of the effective support of the rate function only in terms of the deterministic notion of joint stable length. Finally, as a by-product of our techniques, we deduce some further deterministic results on the asymptotics of a bounded set of isometries. Some of the results in this paper were obtained simultaneously and independently by Boulanger--Mathieu as we discuss in the introduction.