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Permutation Matrices, Wreath Products, and the Distribution of Eigenvalues

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  • Kelly Wieand

    (University of Chicago)

Abstract

We consider a class of random matrix ensembles which can be constructed from the random permutation matrices by replacing the nonzero entries of the n×n permutation matrix matrix with M×M diagonal matrices whose entries are random Kth roots of unity or random points on the unit circle. Let X be the number of eigenvalues lying in a specified arc I of the unit circle, and consider the standardized random variable (X−E[X])/(Var(X))1/2. We show that for a fixed set of arcs I 1,...,I N , the corresponding standardized random variables are jointly normal in the large n limit, and compare the covariance structures which arise with results for other random matrix ensembles.

Suggested Citation

  • Kelly Wieand, 2003. "Permutation Matrices, Wreath Products, and the Distribution of Eigenvalues," Journal of Theoretical Probability, Springer, vol. 16(3), pages 599-623, July.
    • Handle: RePEc:spr:jotpro:v:16:y:2003:i:3:d:10.1023_a:1025616431496
    DOI: 10.1023/A:1025616431496
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    References listed on IDEAS

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    1. 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.
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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.
    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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