in mathematics in general, it is fruitful to allow randomness. indeed, it is often easier to deal with random rather than deterministic objects. it seems miraculous, however, when we are able to say more about deterministic objects by treating them as random ones.

this idea applies in particular to the theory of discrete subgroups of lie groups and locally symmetric manifolds.

the theory of invariant random subgroups (irs), which has been developed quite rapidly during the last decade, has been very fruitful to the study of lattices and their asymptotic invariants. however, restricting to invariant measures limits the scope of problems that one can approach (in particular since the groups involved are highly non-amenable). it was recently realised that the notion of stationary random subgroups (srs) is still very effective and opens paths to deal with questions which were thought to be unreachable.

in the talk i will describe various old and new results concerning arithmetic groups and general locally symmetric manifolds of finite as well as infinite volume that can be proved using `randomness', e.g.:

1. kazhdan-margulis minimal covolume theorem.

2. most hyperbolic manifolds are non-arithmetic (a joint work with a. levit).

3. higher rank manifolds of large volume have a large injectivity radius (joint with abert, bergeron, biringer, nikolov, raimbault and samet).

4. higher rank manifolds of infinite volume have infinite injectivity radius --- conjectured by margulis (joint with m. fraczyk).