On Smooth Mesoscopic Linear Statistics of the Eigenvalues of Random Permutation Matrices

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