  The geometry of multi-parameter families of quantum states is important in
numerous contexts, including adiabatic or nonadiabatic quantum dynamics,
quantum quenches, and the characterization of quantum critical points. Here, we
discuss the Hilbert-space geometry of eigenstates of parameter-dependent
random-matrix ensembles, deriving the full probability distribution of the
quantum geometric tensor for the Gaussian Unitary Ensemble. Our analytical
results give the exact joint distribution function of the Fubini-Study metric
and the Berry curvature. We discuss relations to Levy stable distributions and
compare our results to numerical simulations of random-matrix ensembles as well
as electrons in a random magnetic field.
