Abstract. Smooth linear statistics of random permutation matrices, sampled under a general Ewens distribution, exhibit an interesting nonuniversality 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.

Abstract. We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show that the behavior at different points is independent in the limit and are asymptotically normal. Our methods enables us to study also the wreath product of permutation matrices and diagonal matrices with iid entries and more general class functions on the symmetric group with a multiplicative structure..

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