Random matrices beyond the Cartan classification

Abstract

It is known that hermitean random matrix ensembles can be identified with symmetric coset spaces of Lie groups, or else with tangent spaces of the same. This results in a classification of random matrix ensembles as well as applications in practical calculations of physical observables. In this paper we show that a large number of non-hermitean random matrix ensembles defined by physically motivated symmetries - chiral symmetry, time reversal invariance, space rotation invariance, particle-hole symmetry, or different reality conditions - can likewise be identified with symmetric spaces. We give explicit representations of the random matrix ensembles identified with lateral algebra subspaces, and of the corresponding symmetric subalgebras spanning the group of invariance. Among the ensembles listed we identify as special cases all the hermitean ensembles identified with Cartan classes of symmetric spaces and the three Ginibre ensembles with complex eigenvalues.

41 pages, no figures. References and comments added; the representation of ensemble 15 changed to quaternion real. Version accepted for publication on J. Phys. A

References in corpus (53)

• Random-Matrix Theory of Quantum Transport
• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Random Matrix Theories in Quantum Physics: Common Concepts
• Localization transitions in non-Hermitian quantum mechanics
• Riemannian symmetric superspaces and their origin in random matrix theory
• Random matrix theory and spectral sum rules for the Dirac operator in QCD
• Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering for systems with broken time reversal invariance
• Random matrix model of QCD at finite density and the nature of the quenched limit
• QCD phase diagram and the critical point
• Lattice QCD at Finite Density -- An introductory review
• Non-Hermitean Random Matrix Theory: method of hermitization
• Universal results from an alternate random matrix model for QCD with a baryon chemical potential
• Almost-Hermitian Random Matrices: Crossover from Wigner-Dyson to Ginibre eigenvalue statistics
• Eigenvalue statistics of the real Ginibre ensemble
• gl(N|N) Super-Current Algebras for Disordered Dirac Fermions in Two Dimensions
• Random Matrices close to Hermitian or unitary: overview of methods and results
• Symmetry classes of disordered fermions
• Almost-Hermitian Random Matrices: Eigenvalue Density in the Complex Plane
• Random Matrix Theory of a Chaotic Andreev Quantum Dot
• Probability of Reflection by a Random Laser
• Factorization of Correlation Functions and the Replica Limit of the Toda Lattice Equation
• Systematic Analytical Approach to Correlation Functions of Resonances in Quantum Chaotic Scattering
• Random matrix theory and $QC{D}_3$
• Chiral Symmetry Breaking and the Dirac Spectrum at Nonzero Chemical Potential
• Random matrix theory and symmetric spaces
• Replica field theories, Painleve transcendents, and exact correlation functions
• Eigenvalue correlations in non-Hermitean symplectic random matrices
• On the distribution of transmission eigenvalues in disordered wires
• A Classification of Non-Hermitian Random Matrices
• Parametric Correlations of Scattering Phase Shifts and Fluctuations of Delay Times in Few-Channel Chaotic Scattering
• Effective Lagrangians and Chiral Random Matrix Theory
• Statistics of Real Eigenvalues in Ginibre's Ensemble of Random Real Matrices
• Statistics of S-matrix poles in Few-Channel Chaotic Scattering: Crossover from Isolated to Overlapping Resonances
• Matrix Models and QCD with Chemical Potential
• Fokker-Planck equations and density of states in disordered quantum wires
• A new Chiral Two-Matrix Theory for Dirac Spectra with Imaginary Chemical Potential
• Fermion determinants in matrix models of QCD at nonzero chemical potential
• QCD Dirac operator at nonzero chemical potential: lattice data and matrix model
• Macroscopic Universality : Why QCD in Matter is Subtle?
• The solution of a chiral random matrix model with complex eigenvalues
• The orthogonal ensemble of random matrices and QCD in three dimensions
• Supersymmetric Quenching of the Toda Lattice Equation
• Distribution of complex eigenvalues for symplectic ensembles of non-Hermitian matrices
• Unquenched complex Dirac spectra at nonzero chemical potential: two-colour QCD lattice data versus matrix model
• Signature of wave localisation in the time dependence of a reflected pulse
• Three-dimensional QCD in the adjoint representation and random matrix theory
• A new classification scheme for Random Matrix Theories
• A new universality class describing the insulating regime of disordered wires with strong spin-orbit scattering
• Symmetric space description of carbon nanotubes
• Averages over Ginibre's Ensemble of Random Real Matrices
• Symplectic Structure of the Real Ginibre Ensemble
• Random Matrix Theory at Nonzero $\mu$ and $T$
• Lectures on random matrix theory and symmetric spaces

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• Products of Independent Gaussian Random Matrices
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