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Non-Hermitian extensions of Wishart random matrix ensembles
We briefly review the solution of three ensembles of non-Hermitian random
matrices generalizing the Wishart-Laguerre (also called chiral) ensembles.
These generalizations are realized as Gaussian two-matrix models, where the
complex eigenvalues of the product of the two independent rectangular matrices
are sought, with the matrix elements of both matrices being either real,
complex or quaternion real. We also present the more general case depending on
a non-Hermiticity parameter, that allows us to interpolate between the
corresponding three Hermitian Wishart ensembles with real eigenvalues and the
maximally non-Hermitian case. All three symmetry classes are explicitly solved
for finite matrix size NxM for all complex eigenvalue correlations functions
(and real or mixed correlations for real matrix elements). These are given in
terms of the corresponding kernels built from orthogonal or skew-orthogonal
Laguerre polynomials in the complex plane. We then present the corresponding
three Bessel kernels in the complex plane in the microscopic large-N scaling
limit at the origin, both at weak and strong non-Hermiticity with M-N greater
or equal to 0 fixed.Comment: 21 pages, 3 figures, presented at "Random Matrices, Statistical
Physics and Information Theory", Krakow, 26-30/09/201
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