  The term "strong approximation" is used to describe phenomena where an
arithmetic group as well as all of its Zariski dense subgroups have a large
image in the congruence quotients. We exhibit analogues of such phenomena in a
probabilistic, rather than arithmetic, setting.
  Let T be the binary rooted tree, Aut(T) its automorphism group. To a given
m-tuple a = {a_1,a_2,...,a_m} in Aut(T), we associate a tower of 2m-regular
Schreier graphs ...X_n-->X_{n-1}-->...-->X_0. The vertices of X_n are the
n^{th} level of the tree and two such are connected by an edge if a generator
takes one to the other.
  When {a_i} are independent Haar-random elements of Aut(T) we retrieve the
standard model for iterated random 2-lifts studied, for example by Bilu-Linial.
If w={w_1,w_2,...,w_l} are words in the free group F_m, the random
substitutions w(a) := {w_1(a),...,w_l(a)} give rise to new models for random
towers of 2l-regular graphs: ...Y_n-->Y_{n-1}-->...-->Y_0.
  With the above notation, the following hold almost surely, for every non
cyclic subgroup D in F_m: (i) the graphs $Y_n$ have a bounded number of
connected components, (ii) these connected components form a family of expander
graphs, (iii) the closure of D has positive Hausdorff dimension as a subgroup
of the (metric) group Aut(T).
