# The Complex Laguerre Symplectic Ensemble of Non-Hermitian Matrices

Akemann G (2005) *Nucl.Phys. B* 730(3): 253-299.

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We solve the complex extension of the chiral Gaussian Symplectic Ensemble,defined as a Gaussian two-matrix model of chiral non-Hermitian quaternion realmatrices. This leads to the appearance of Laguerre polynomials in the complexplane and we prove their orthogonality. Alternatively, a complex eigenvaluerepresentation of this ensemble is given for general weight functions. Allk-point correlation functions of complex eigenvalues are given in terms of thecorresponding skew orthogonal polynomials in the complex plane for finite-N,where N is the matrix size or number of eigenvalues, respectively. We alsoallow for an arbitrary number of complex conjugate pairs of characteristicpolynomials in the weight function, corresponding to massive quark flavours inapplications to field theory. Explicit expressions are given in the large-Nlimit at both weak and strong non-Hermiticity for the weight of the Gaussiantwo-matrix model. This model can be mapped to the complex Dirac operatorspectrum with non-vanishing chemical potential. It belongs to the symmetryclass of either the adjoint representation or two colours in the fundamentalrepresentation using staggered lattice fermions.

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Akemann G. The Complex Laguerre Symplectic Ensemble of Non-Hermitian Matrices.

*Nucl.Phys. B*. 2005;730(3):253-299.Akemann, G. (2005). The Complex Laguerre Symplectic Ensemble of Non-Hermitian Matrices.

*Nucl.Phys. B*,*730*(3), 253-299. doi:10.1016/j.nuclphysb.2005.09.039Akemann, G. (2005). The Complex Laguerre Symplectic Ensemble of Non-Hermitian Matrices.

*Nucl.Phys. B*730, 253-299.Akemann, G., 2005. The Complex Laguerre Symplectic Ensemble of Non-Hermitian Matrices.

*Nucl.Phys. B*, 730(3), p 253-299. G. Akemann, “The Complex Laguerre Symplectic Ensemble of Non-Hermitian Matrices”,

*Nucl.Phys. B*, vol. 730, 2005, pp. 253-299. Akemann, G.: The Complex Laguerre Symplectic Ensemble of Non-Hermitian Matrices. Nucl.Phys. B. 730, 253-299 (2005).

Akemann, Gernot. “The Complex Laguerre Symplectic Ensemble of Non-Hermitian Matrices”.

*Nucl.Phys. B*730.3 (2005): 253-299.
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arXiv hep-th/0507156