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?