On absolute moments of characteristic polynomials of a certain class of complex random matrices

Abstract

Integer moments of the spectral determinant $|\det (zI-W){|}^2$ of complex random matrices $W$ are obtained in terms of the characteristic polynomial of the Hermitian matrix $W{W}^{\ast }$ for the class of matrices $W=AU$ where $A$ is a given matrix and $U$ is random unitary. This work is motivated by studies of complex eigenvalues of random matrices and potential applications of the obtained results in this context are discussed.

arXiv comment41 page, typos corrected

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• Derivation of an eigenvalue probability density function relating to the Poincare disk
• Unveiling the significance of eigenvectors in diffusing non-hermitian matrices by identifying the underlying Burgers dynamics
• Characteristic polynomials of complex random matrices and Painlevé transcendents
• Maximum of the characteristic polynomial of the Ginibre ensemble
• Matrix averages relating to the Ginibre ensembles
• On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values
• Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates
• Matrix models and eigenvalue statistics for truncations of classical ensembles of random unitary matrices
• Bulk Universality for Complex non-Hermitian Matrices with Independent and Identically Distributed Entries
• Schur function averages for the real Ginibre ensemble
• Extreme eigenvalues and the emerging outlier in rank-one non-Hermitian deformations of the Gaussian Unitary Ensemble
• On the correlation functions of the characteristic polynomials of the sparse hermitian random matrices
• Skew orthogonal polynomials for the real and quaternion real Ginibre ensembles and generalizations
• Wilson loops and free energies in $3d$ $\mathcal{N}=4$ SYM: exact results, exponential asymptotics and duality
• Generalised unitary group integrals of Ingham-Siegel and Fisher-Hartwig type