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On Smooth Mesoscopic Linear Statistics of the Eigenvalues of Random Permutation Matrices

Author

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  • Valentin Bahier

    (University of Bristol School of Mathematics)

  • Joseph Najnudel

    (University of Bristol School of Mathematics)

Abstract

We study the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group. If we apply a smooth enough test function f to all the determinations of the eigenangles of the permutations, we get a convergence in distribution when the order of the permutation tends to infinity. Two distinct kinds of limit appear: if $$f(0)\ne 0$$ f ( 0 ) ≠ 0 , we have a central limit theorem with a logarithmic variance; and if $$f(0) = 0$$ f ( 0 ) = 0 , the convergence holds without normalization and the limit involves a scale-invariant Poisson point process.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:3:d:10.1007_s10959-021-01106-4
    DOI: 10.1007/s10959-021-01106-4
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    References listed on IDEAS

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    1. Kelly Wieand, 2003. "Permutation Matrices, Wreath Products, and the Distribution of Eigenvalues," Journal of Theoretical Probability, Springer, vol. 16(3), pages 599-623, July.
    2. 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.
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    1. 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.

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