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Journal of Theoretical Probability

, Volume 32, Issue 2, pp 974–1022 | Cite as

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

  • Valentin BahierEmail author
Article
  • 40 Downloads

Abstract

We are interested in two random matrix ensembles related to permutations: the ensemble of permutation matrices following Ewens’ distribution of a given parameter \(\theta >0\) 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 elements 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.

Keywords

Random matrices Permutations Distribution of eigenvalues Wreath product Central limit theorem 

Mathematics Subject Classification

60B20 60F05 20B30 

Notes

Acknowledgements

The author wishes to thank his Ph.D. advisor Joseph Najnudel for suggesting the problem and helpful comments.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institut de Mathématiques de ToulouseCedex 9 ToulouseFrance

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