Asymptotic Freeness of Random Permutation Matrices from Gaussian Matrices

It is shown that an independent family of uniformly distributed random permutation matrices is asymptotically ∗-free from an independent family of square complex Gaussian matrices and from an independent family of complexWishart matrices, and that in both cases the convergence in ∗-distribution actually holds almost surely.

An immediate consequence is that, if the rows of a GUE matrix are randomly permuted, then the resulting (non self-adjoint) random matrix has a ∗-distribution which is asymptotically circular; similarly, a random permutation of the rows of a complex Wishart matrix results in a random matrix which is asymptotically ∗-distributed like an R-diagonal element from free probability theory.