  We consider the variance of sums of arithmetic functions over random short
intervals in the function field setting. Based on the analogy between
factorizations of random elements of $\mathbb{F}_q[T]$ into primes and the
factorizations of random permutations into cycles, we give a simple but general
formula for these variances in the large $q$ limit for arithmetic functions
that depend only upon factorization structure. From this we derive new
estimates, quickly recover some that are already known, and make new
conjectures in the setting of the integers.
  In particular we make the combinatorial observation that any function of this
sort can be decomposed into a sum of functions $u$ and $v$, depending on the
size of the short interval, with $u$ making a negligible contribution to the
variance, and $v$ asymptotically contributing diagonal terms only.
  This variance evaluation is closely related to the appearance of random
matrix statistics in the zeros of families of L-functions and sheds light on
the arithmetic meaning of this phenomenon.
