On fluctuations of eigenvalues of random permutation matrices

TL;DR

This paper investigates the fluctuation behavior of eigenvalues in random permutation matrices under Ewens distribution, revealing non-Gaussian limits for smooth statistics and Gaussian limits for less smooth ones.

Contribution

It uncovers a non-universality phenomenon in the fluctuation distributions, showing how smoothness affects the asymptotic Gaussianity or non-Gaussianity of eigenvalue statistics.

Findings

Smooth statistics have non-Gaussian, infinitely divisible fluctuations.

Less smooth statistics exhibit Gaussian fluctuations with diverging variance.

Fluctuation behavior depends on the smoothness of the observable function.

Abstract

Smooth linear statistics of random permutation matrices, sampled under a general Ewens distribution, exhibit an interesting non-universality phenomenon. Though they have bounded variance, their fluctuations are asymptotically non-Gaussian but infinitely divisible. The fluctuations are asymptotically Gaussian for less smooth linear statistics for which the variance diverges. The degree of smoothness is measured in terms of the quality of the trapezoidal approximations of the integral of the observable.