I will discuss a quantitative variant of the classical Kazhdan-Margulis theorem generalized to stationary actions of semisimple groups over local fields. More precisely, the probability that the stabilizer of a random point admits a non-trivial intersection with a small r-neighborhood of the identity is at most a constant times r^d, for some explicit d > 0 depending only on the semisimple group in question. Our proof involves some of the original ideas of Kazhdan and Margulis, combined with methods of Margulis functions as well as (C,alpha)-good functions on varieties. As an application, we present a new unified proof of the fact that all lattices in these groups are weakly cocompact, i.e admit a spectral gap. The talk is based on a preprint joint with Gelander and Margulis.