Characteristic polynomials in real Ginibre ensembles

Abstract

We calculate the average of two characteristic polynomials for the real Ginibre ensemble of asymmetric random matrices, and its chiral counterpart. Considered as quadratic forms they determine a skew-symmetric kernel from which all complex eigenvalue correlations can be derived. Our results are obtained in a very simple fashion without going to an eigenvalue representation, and are completely new in the chiral case. They hold for Gaussian ensembles which are partly symmetric, with kernels given in terms of Hermite and Laguerre polynomials respectively, depending on an asymmetry parameter. This allows us to interpolate between the maximally asymmetric real Ginibre and the Gaussian Orthogonal Ensemble, as well as their chiral counterparts.

9 pages; v2 typos corrected

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• Truncations of Random Orthogonal Matrices
• Chiral random matrix theory for two-color QCD at high density
• Products of Independent Quaternion Ginibre Matrices and their Correlation Functions
• Gap Probabilities in Non-Hermitian Random Matrix Theory
• A new approach to derive Pfaffian structures for random matrix ensembles
• Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble
• Asymmetric correlation matrices: an analysis of financial data
• Matrix averages relating to the Ginibre ensembles
• Mixing of orthogonal and skew-orthogonal polynomials and its relation to Wilson RMT
• The chiral Gaussian two-matrix ensemble of real asymmetric matrices
• The limiting Kac random polynomial and truncated random orthogonal matrices
• Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles
• Lyapunov exponents for products of rectangular real, complex and quaternionic Ginibre matrices
• Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
• Schur function averages for the real Ginibre ensemble
• Diffusion method in Random Matrix Theory
• Characteristic polynomials of products of Wigner matrices: finite-N results and Lyapunov universality
• Schur expansion of random-matrix reproducing kernels