  We obtain sharp estimates on the growth rate of stable commutator length on
random (geodesic) words, and on random walks, in hyperbolic groups and groups
acting nondegenerately on hyperbolic spaces. In either case, we show that with
high probability stable commutator length of an element of length $n$ is of
order $n/\log{n}$.
  This establishes quantitative refinements of qualitative results of
Bestvina-Fujiwara and others on the infinite dimensionality of 2-dimensional
bounded cohomology in groups acting suitably on hyperbolic spaces, in the sense
that we can control the geometry of the unit balls in these normed vector
spaces (or rather, in random subspaces of their normed duals).
  As a corollary of our methods, we show that an element obtained by random
walk of length $n$ in a mapping class group cannot be written as a product of
fewer than $O(n/\log{n})$ reducible elements, with probability going to 1 as
$n$ goes to infinity. We also show that the translation length on the complex
of free factors of a random walk of length $n$ on the outer automorphism group
of a free group grows linearly in $n$.

Group Theory;Dynamical Systems;Geometric Topology