A {\em permutation statistic} is a complex-valued function on the symmetric group $\mathfrak{S}_n$.  We describe a notion of `locality' which measures the complexity of permutation statistics. Applications are given to the asymptotic behavior of families of statistics as the parameter $n$ grows. The key technical tool is an irreducible character evaluation on the symmetric group algebra which involves a novel combinatorics of `monotonic ribbon tableaux'. Joint with Zach Hamaker.