I will give a background about acylindrical actions, hyperbolic spaces and acylindrically hyperbolic groups. I will especially highlight some of the existing theory as it relates to the behavior of "random" elements and "random" subgroups of such groups (i.e. those obtained by random walks on the group). The question I am interested in (in joint ongoing work with T. Fernos) is how much of this behavior generalizes to acylindrical actions on products of hyperbolic spaces. The motivation behind this question is the behavior of Higher rank lattices, Mapping class groups and decomposable RAAGs. I will talk about some of the result we have obtained in this direction.