Point count statistics for families of curves over finite fields

We investigate the distribution of the number of $F_p$-points of curves in various families,
where $F_p$ is the finite field with $p$ elements.  If we consider a family of curves having
fixed genus $g$ and let $p$ tend to infinity the situation is fairly well understood -
the distribution of the point count fluctuations are given by generalized Sato-Tate distributions,
which in turn is closely related to random matrix theory.  On the other hand, if $p$ is fixed
and we let $g$ tend to infinity (or taking $p,g$ to infinity in some arbitrary way), the situation
is less clear, e.g., since the number of points on a curve cannot be negative, the random
matrix theory model is not valid in this setting.  However, for some families of curves,
certain "coin flip models" (closely related to character sums in the case of hyperelliptic curves)
can be used to describe the fluctuations; using this we can show that the point count fluctuations
are Gaussian in the large genus limit