Freeness and the transposes of unitarily invariant random matrices

Abstract

We show that real second order freeness appears in the study of Haar unitary and unitarily invariant random matrices when transposes are also considered. In particular we obtain the unexpected result that a unitarily invariant random matrix will be asymptotically free from its transpose.

Bipartite unitary gates and billiard dynamics in the Weyl chamber

“…TrA k (B † ) m , vanish in the limit N → ∞. A recent unexpected result of Mingo and Popa [35] concerning Haar random unitary matrices states that U and U T are asymptotically free, and in order to substantiate our interpretation of the convergence to the Marchenko-Pastur distribution in Appendix A we compute numerically the averaged traces of products of two random matrices for the cases of interest discussed throughout our paper. Our results allow us to advance the following Conjecture 3.…”

Freeness and The Partial Transposes of Wishart Random Matrices

“…For many eigenvalue results there is no distinction between the real and complex case. In [9] we showed that when it comes to freeness there is a difference, in particular with respect to the behaviour of the transpose. In this paper we show that with the partial transpose we continue to see a difference between the real and complex cases.…”

“…Since W is unitarily invariant, a consequence of the results from [9] is that W and W T are asymptotically free if d 1 d 2 → ∞. In this section we will present the main results of the paper, which, using the relation form Theorem 9, gives an improvement of the result mentioned above.…”

“…She showed that the fluctuation moments of real Gaussian and Wishart matrices require the transpose to taken into account. Later in [9] the authors showed that the transpose can also appear at the first order level. Namely, that for many complex ensembles a random matrix could be asymptotically free from its transpose.…”

An asymptotic property of large matrices with identically distributed Boolean independent entries

“…i l i l+1 : l ∈ B} satisfy the conditions from Theorem 3.4. Henceforth, if σ ∧ ω( − → k ) is not − → ξ -alternating, equation (7) gives…”