  We study the limiting behavior of smooth linear statistics of the spectrum of
random permutation matrices in the mesoscopic regime, when the permutation
follows one of the Ewens measures on the symmetric group. If we apply a smooth
enough test function $f$ to all the determinations of the eigenangles of the
permutations, we get a convergence in distribution when the order of the
permutation tends to infinity. Two distinct kinds of limit appear: if $f(0)\neq
0$, we have a central limit theorem with a logarithmic variance, and if $f(0) =
0$, the convergence holds without normalization and the limit involves a
scale-invariant Poisson point process.
