Étale homological stability and arithmetic statistics

We relate asymptotic point counts and arithmetic statistics over finite fields to homological stability and representation stability over
$\mathbb{C}$
in the example of configuration spaces of n points on smooth varieties. In order to accomplish this, we establish subexponential bounds on the growth of the unstable cohomology of such spaces. We apply this and étale homological stability to compute the large n limits of various arithmetic statistics of configuration spaces of varieties over
$\mathbb {F}_q$
.