We are interested in two random matrix ensembles related to permutations: the ensemble of permutation matrices following Ewens’ distribution of a given param-eter θ > 0 and its modification where entries equal to 1 in the matrices are replaced by independent random variables uniformly distributed on the unit circle. For the ele-ments of each ensemble, we focus on the random numbers of eigenvalues lying in some specified arcs of the unit circle. We show that for a finite number of fixed arcs, the fluctuation of the numbers of eigenvalues belonging to them is asymptotically Gaussian. Moreover, for a single arc, we extend this result to the case where the length goes to zero sufficiently slowly when the size of the matrix goes to infinity. Finally, we investigate the behavior of the largest and smallest spacings between two distinct consecutive eigenvalues.

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• On the Number of Eigenvalues of Modified Permutation Matrices in Mesoscopic Intervals