Asymptotic Freeness of Random Permutation Matrices from Gaussian Matrices

Abstract

We show 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 complex Wishart 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.