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Asymptotic Freeness of Random Permutation Matrices from Gaussian Matrices

Mihail G. Neagu
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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.

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Mihail G. Neagu. 2004-12-03. Asymptotic Freeness of Random Permutation Matrices from Gaussian Matrices. https://arxiv.org/abs/math/0410028

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