  Consider the number of integers in a short interval that can be represented
as a sum of two squares. What is an estimate for the variance of these counts
over random short intervals? We resolve a function field variant of this
problem in the large $q$ limit, finding a connection to the $z$-measures first
investigated in the context of harmonic analysis on the infinite symmetric
group. A similar connection to $z$-measures is established for sums over short
intervals of the divisor functions $d_z(n)$. We use these results to make
conjectures in the setting of the integers which match very well with
numerically produced data. Our proofs depend on equidistribution results of N.
Katz and W. Sawin.
