# The chiral Gaussian two-matrix ensemble of real asymmetric matrices

Akemann G, Phillips MJ, Sommers H-J (2009)

J.Phys.A 43(8).

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Abstract

We solve a family of Gaussian two-matrix models with rectangular Nx(N+v)matrices, having real asymmetric matrix elements and depending on anon-Hermiticity parameter mu. Our model can be thought of as the chiralextension of the real Ginibre ensemble, relevant for Dirac operators in thesame symmetry class. It has the property that its eigenvalues are either real,purely imaginary, or come in complex conjugate eigenvalue pairs. The eigenvaluejoint probability distribution for our model is explicitly computed, leading toa non-Gaussian distribution including K-Bessel functions. All n-point densitycorrelation functions are expressed for finite N in terms of a Pfaffian form.This contains a kernel involving Laguerre polynomials in the complex plane as abuilding block which was previously computed by the authors. This kernel can beexpressed in terms of the kernel for complex non-Hermitian matrices,generalising the known relation among ensembles of Hermitian random matrices.Compact expressions are given for the density at finite N as an example, aswell as its microscopic large-N limits at the origin for fixed v at strong andweak non-Hermiticity.

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Akemann G, Phillips MJ, Sommers H-J. The chiral Gaussian two-matrix ensemble of real asymmetric matrices.

*J.Phys.A*. 2009;43(8).Akemann, G., Phillips, M. J., & Sommers, H. - J. (2009). The chiral Gaussian two-matrix ensemble of real asymmetric matrices.

*J.Phys.A*,*43*(8).Akemann, G., Phillips, M. J., and Sommers, H. - J. (2009). The chiral Gaussian two-matrix ensemble of real asymmetric matrices.

*J.Phys.A*43.Akemann, G., Phillips, M.J., & Sommers, H.-J., 2009. The chiral Gaussian two-matrix ensemble of real asymmetric matrices.

*J.Phys.A*, 43(8).G. Akemann, M.J. Phillips, and H.-J. Sommers, “The chiral Gaussian two-matrix ensemble of real asymmetric matrices”,

*J.Phys.A*, vol. 43, 2009.Akemann, G., Phillips, M.J., Sommers, H.-J.: The chiral Gaussian two-matrix ensemble of real asymmetric matrices. J.Phys.A. 43, (2009).

Akemann, Gernot, Phillips, M. J., and Sommers, H. -J. “The chiral Gaussian two-matrix ensemble of real asymmetric matrices”.

*J.Phys.A*43.8 (2009).
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