Permutation matrices, wreath products, and the distribution of eigenvalues

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 x n permutation matrix matrix with M x 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.