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

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  • Valentin Bahier

    (Institut de Mathématiques de Toulouse)

Abstract

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

Suggested Citation

  • Valentin Bahier, 2019. "On the Number of Eigenvalues of Modified Permutation Matrices in Mesoscopic Intervals," Journal of Theoretical Probability, Springer, vol. 32(2), pages 974-1022, June.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:2:d:10.1007_s10959-017-0798-5
    DOI: 10.1007/s10959-017-0798-5
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    References listed on IDEAS

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    1. Dang, K. & Zeindler, D., 2014. "The characteristic polynomial of a random permutation matrix at different points," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 411-439.
    2. Dirk Zeindler, 2013. "Central Limit Theorem for Multiplicative Class Functions on the Symmetric Group," Journal of Theoretical Probability, Springer, vol. 26(4), pages 968-996, December.
    3. Hambly, B. M. & Keevash, P. & O'Connell, N. & Stark, D., 2000. "The characteristic polynomial of a random permutation matrix," Stochastic Processes and their Applications, Elsevier, vol. 90(2), pages 335-346, December.
    4. Kelly Wieand, 2003. "Permutation Matrices, Wreath Products, and the Distribution of Eigenvalues," Journal of Theoretical Probability, Springer, vol. 16(3), pages 599-623, July.
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    Cited by:

    1. Valentin Bahier & Joseph Najnudel, 2022. "On Smooth Mesoscopic Linear Statistics of the Eigenvalues of Random Permutation Matrices," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1640-1661, September.

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