Complex symmetric, self-dual, and Ginibre random matrices: Analytical results for three classes of bulk and edge statistics

Abstract

Recently, a conjecture about the local bulk statistics of complex eigenvalues has been made based on numerics. It claims that there are only three universality classes, which have all been observed in open chaotic quantum systems. Motivated by these new insights, we compute and compare the expectation values of $k$ pairs of complex conjugate characteristic polynomials in three ensembles of Gaussian non-Hermitian random matrices representative for the three classes: the well-known complex Ginibre ensemble, complex symmetric and complex self-dual matrices. In the Cartan classification scheme of non-Hermitian random matrices they are labelled as class A, AI${}^{†}$ and AII${}^{†}$, respectively. Using the technique of Grassmann variables, we derive explicit expressions for a single pair of expected characteristic polynomials for finite as well as infinite matrix dimension. For the latter we consider the global limit as well as zoom into the edge and the bulk of the spectrum, providing new analytical results for classes AI${}^{†}$ and AII${}^{†}$. For general $k$, we derive the effective Lagrangians corresponding to the non-linear $\sigma$-models in the respective physical systems. Interestingly, they agree for all three ensembles, while the corresponding Goldstone manifolds, over which one has to perform the remaining integrations, are different and equal the three classical compact groups in the bulk. In particular, our analytical results show that these three ensembles have indeed different local bulk and edge spectral statistics, corroborating the conjecture further.

47 pages, 2 figures, v2: typos corrected and minor clarifications added

References in corpus (37)

• Non-Hermitian Physics
• Symmetry and Topology in Non-Hermitian Physics
• Universal results from an alternate random matrix model for QCD with a baryon chemical potential
• Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos
• The Ginibre ensemble of real random matrices and its scaling limits
• Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems
• Random matrices: Universality of local spectral statistics of non-Hermitian matrices
• Universality classes of non-Hermitian random matrices
• Non-Hermitian Random Matrix Theory and Lattice QCD with Chemical Potential
• Factorization of Correlation Functions and the Replica Limit of the Toda Lattice Equation
• Characteristic Polynomials of Complex Random Matrix Models
• Random Quantum Operations
• Microscopic Spectrum of the Wilson Dirac Operator
• The Complex Laguerre Symplectic Ensemble of Non-Hermitian Matrices
• Random matrices beyond the Cartan classification
• Derivation of determinantal structures for random matrix ensembles in a new way
• Spectral transitions and universal steady states in random Kraus maps and circuits
• Massive partition functions and complex eigenvalue correlations in Matrix Models with symplectic symmetry
• From integrability to chaos in quantum Liouvillians
• Ratios of characteristic polynomials in complex matrix models
• Replica treatment of non-Hermitian disordered Hamiltonians
• A new approach to derive Pfaffian structures for random matrix ensembles
• Singular Behavior At The Edge of Laughlin States
• Characteristic polynomials of complex random matrices and Painlevé transcendents
• Spectral Properties of the Wilson Dirac Operator and random matrix theory
• Mixing of orthogonal and skew-orthogonal polynomials and its relation to Wilson RMT
• The boundary density profile of a Coulomb droplet. Freezing at the edge
• The chiral Gaussian two-matrix ensemble of real asymmetric matrices
• New universality classes of the non-Hermitian Dirac operator in QCD-like theories
• Spectra of large time-lagged correlation matrices from Random Matrix Theory
• Biorthogonal Polynomials for Potentials of two Variables and External Sources at the Denominator
• Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles
• Spacing distribution in the 2D Coulomb gas: Surmise and symmetry classes of non-Hermitian random matrices at non-integer $\beta$
• Shift of Symmetries of Naive Fermions in QCD-like Lattice Theories
• Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
• On the Correlation Functions of the Characteristic Polynomials of Non-Hermitian Random Matrices with Independent Entries
• Two transitions in complex eigenvalue statistics: Hermiticity and integrability breaking

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