  We continue the study of a general class of spaces of 0-cycles on a manifold
defined and begun by Farb-Wolfson-Wood. Using work of Gadish on linear subspace
arrangements, we obtain representation stability for the cohomology of the
ordered version of these spaces. We establish subexponential bounds on the
growth of unstable cohomology, and the Grothendieck-Lefschetz trace formula
then allows us to translate these topological stability phenomena to
stabilization of statistics for spaces of 0-cycles over finite fields. In
particular, we show that the average value of certain arithmetic quantities
associated to rational maps over finite fields stabilizes as the degree goes to
infinity.
