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We obtain sharp estimates on the growth rate of stable commutator length onrandom (geodesic) words, and on random walks, in hyperbolic groups and groupsacting nondegenerately on hyperbolic spaces. In either case, we show that withhigh probability stable commutator length of an element of length $n$ is oforder $n/\log{n}$.  This establishes quantitative refinements of qualitative results ofBestvina-Fujiwara and others on the infinite dimensionality of 2-dimensionalbounded cohomology in groups acting suitably on hyperbolic spaces, in the sensethat we can control the geometry of the unit balls in these normed vectorspaces (or rather, in random subspaces of their normed duals).  As a corollary of our methods, we show that an element obtained by randomwalk of length $n$ in a mapping class group cannot be written as a product offewer 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 complexof free factors of a random walk of length $n$ on the outer automorphism groupof a free group grows linearly in $n$.

Statistics and compression of scl