On fluctuations of eigenvalues of random permutation matrices
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 620-647.

Les statistiques linéaires d’observables régulières du spectre de matrices de permutations, choisies aléatoirement sous une distribution générale de Ewens, donnent lieu à un phénomène intéressant de non-universalité. Bien qu’elles aient une variance bornée, leurs fluctuations ne sont pas asymptotiquement Gaussiennes, mais infiniment divisibles. Si l’observable est moins régulière, la variance diverge et les fluctuations sont Gaussiennes. Le degré de régularité est mesuré en termes de la qualité de l’approximation trapézoidale de l’intégrale de l’observable.

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.

DOI : 10.1214/13-AIHP569
Classification : 60F05, 15B52, 60B20, 60B15, 60C05, 60E07, 65D30
Mots-clés : random matrices, linear eigenvalue statistics, random permutations, infinitely divisible distributions, trapezoidal approximations 
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Ben Arous, Gérard; Dang, Kim. On fluctuations of eigenvalues of random permutation matrices. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 620-647. doi : 10.1214/13-AIHP569. http://www.numdam.org/articles/10.1214/13-AIHP569/

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