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The Dirichlet Markov Ensemble

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  • Chafaï, Djalil

Abstract

We equip the polytope of nxn Markov matrices with the normalized trace of the Lebesgue measure of . This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean (1/n,...,1/n). We show that if is such a random matrix, then the empirical distribution built from the singular values of tends as n-->[infinity] to a Wigner quarter-circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of tends as n-->[infinity] to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of is of order when n is large.

Suggested Citation

  • Chafaï, Djalil, 2010. "The Dirichlet Markov Ensemble," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 555-567, March.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:3:p:555-567
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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.
    2. G. Goldberg & P. Okunev & M. Neumann & H. Schneider, 2000. "Distribution of Subdominant Eigenvalues of Random Matrices," Methodology and Computing in Applied Probability, Springer, vol. 2(2), pages 137-151, August.
    3. Edelman, Alan, 1997. "The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law," Journal of Multivariate Analysis, Elsevier, vol. 60(2), pages 203-232, February.
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