The chiral Gaussian two-matrix ensemble of real asymmetric matrices

Abstract

We solve a family of Gaussian two-matrix models with rectangular Nx(N+v) matrices, having real asymmetric matrix elements and depending on a non-Hermiticity parameter mu. Our model can be thought of as the chiral extension of the real Ginibre ensemble, relevant for Dirac operators in the same symmetry class. It has the property that its eigenvalues are either real, purely imaginary, or come in complex conjugate eigenvalue pairs. The eigenvalue joint probability distribution for our model is explicitly computed, leading to a non-Gaussian distribution including K-Bessel functions. All n-point density correlation functions are expressed for finite N in terms of a Pfaffian form. This contains a kernel involving Laguerre polynomials in the complex plane as a building block which was previously computed by the authors. This kernel can be expressed 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, as well as its microscopic large-N limits at the origin for fixed v at strong and weak non-Hermiticity.

31 pages, 6 figures

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• Matrix Models and QCD with Chemical Potential
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• Entangled random pure states with orthogonal symmetry: exact results
• Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
• Spectral density of a Wishart model for nonsymmetric Correlation Matrices
• Relativistic Cooper pairing in the microscopic limit of chiral random matrix theory
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• Nonrelativistic Banks-Casher relation and random matrix theory for multi-component fermionic superfluids
• Complex symmetric, self-dual, and Ginibre random matrices: Analytical results for three classes of bulk and edge statistics