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Let $A$ be a uniformly random $k\times k$ permutation matrix, and $A_1,\ldots, A_m$ be the $m$ independent copies of $A$. Here the uniform distribution is with respect to the $k!$ possible permutation matrices.

Question: Up to proper normalization factors involving $m$, what is the asymptotic distribution of $\sum_{i=1}^m A_i - \mathbf{E}(A)$ and its eigenvalues? Can we expect some kind of Gaussian approximation and eigenvalue laws?

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  • $\begingroup$ It obviously satisfies a CLT, with the only nuisance being that the covariance matrix is singular (rank $(n-1)^2$). If you ignore the last row and last column, which can be determined from the others, any statement of the multidimensional CLT, Edgeworth expansion (lattice version), Berry-Esseen, etc etc, you can find will apply and you can easily compute the parameters. Hopefully someone else will answer about eigenvalues. $\endgroup$ Oct 7, 2016 at 2:09
  • $\begingroup$ Asymptotically, the empirical measure of eigenvalues should behave like a sum of $m$ Haar unitaries, in the limit when $m$ is fixed and $k\to \infty$. The limit for $m$ unitaries was computed in the following paper of Basak and Dembo: projecteuclid.org/euclid.ecp/1465315608 . Justifying this is however a challenging problem. Some partial results are available in various regimes, not all written up.... $\endgroup$ Oct 7, 2016 at 4:45

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