Random Matrices close to Hermitian or unitary: overview of methods and results

Abstract

The paper discusses progress in understanding statistical properties of complex eigenvalues (and corresponding eigenvectors) of weakly non-unitary and non-Hermitian random matrices. Ensembles of this type emerge in various physical contexts, most importantly in random matrix description of quantum chaotic scattering as well as in the context of QCD-inspired random matrix models.

Published version, with a few more misprints corrected

References in corpus (3)

• Quantum Graphs: A simple model for Chaotic Scattering
• Large scale correlations in normal and general non-Hermitian matrix ensembles
• Microscopic universality of complex matrix model correlation functions at weak non-Hermiticity

Cited by in corpus (101)

• Non-Hermitian Physics
• Random Matrices and Chaos in Nuclear Physics: Nuclear Reactions
• Universal results from an alternate random matrix model for QCD with a baryon chemical potential
• Scattering, reflection and impedance of waves in chaotic and disordered systems with absorption
• The Ginibre ensemble of real random matrices and its scaling limits
• Super-Radiant Dynamics, Doorways, and Resonances in Nuclei and Other Open Mesoscopic Systems
• Factorization of Correlation Functions and the Replica Limit of the Toda Lattice Equation
• Symmetry Classification and Universality in Non-Hermitian Many-Body Quantum Chaos by the Sachdev-Ye-Kitaev Model
• Statistics of resonance width shifts as a signature of eigenfunction non-orthogonality
• Symmetry classes of open fermionic quantum matter
• Quantum conductance problems and the Jacobi ensemble
• Semiclassical structure of chaotic resonance eigenfunctions
• Spectral gaps and mid-gap states in random quantum master equations
• Statistics of Resonances and Delay Times in Random Media: Beyond Random Matrix Theory
• Non-Hermitian Euclidean random matrix theory
• The Complex Laguerre Symplectic Ensemble of Non-Hermitian Matrices
• Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach
• Statistics of Real Eigenvalues in Ginibre's Ensemble of Random Real Matrices
• Spectral Theory of Sparse Non-Hermitian Random Matrices
• Matrix Models and QCD with Chemical Potential
• Integrable Structure of Ginibre's Ensemble of Real Random Matrices and a Pfaffian Integration Theorem
• The distribution of overlaps between eigenvectors of Ginibre matrices
• Random matrices beyond the Cartan classification
• On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry
• Induced vs Spontaneous Breakdown of S-matrix Unitarity: Probability of No Return in Quantum Chaotic and Disordered Systems
• Gaussian fluctuations for non-Hermitian random matrix ensembles
• Out-of-time-ordered correlator in the quantum bakers map and truncated unitary matrices
• Resonances in open quantum maps
• Truncations of Random Orthogonal Matrices
• From closed to open 1D Anderson model: Transport versus spectral statistics
• Massive partition functions and complex eigenvalue correlations in Matrix Models with symplectic symmetry
• Delocalization of eigenvectors of random matrices with independent entries
• Universality at weak and strong non-Hermiticity beyond the elliptic Ginibre ensemble
• Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensemble
• On absolute moments of characteristic polynomials of a certain class of complex random matrices
• A few remarks on Colour-Flavour Transformations,truncations of random unitary matrices, Berezin reproducing kernels and Selberg type integrals
• Gap Probabilities in Non-Hermitian Random Matrix Theory
• Large deviations of radial statistics in the two-dimensional one-component plasma
• The fate of reentrant localization phenomenon in the one-dimensional dimerized quasiperiodic chain with long-range hopping
• Ratios of characteristic polynomials in complex matrix models
• Analogies between random matrix ensembles and the one-component plasma in two-dimensions
• Inhomogeneous losses and complexness of wave functions in chaotic cavities
• Statistics of eigenfunctions in open chaotic systems: a perturbative approach
• Characteristic polynomials of complex random matrices and Painlevé transcendents
• Non-Hermitean Random Matrix Theory: Summation of Planar Diagrams, the "Single-Ring" Theorem and the Disk-Annulus Phase Transition
• Universality and its limits in non-Hermitian many-body quantum chaos using the Sachdev-Ye-Kitaev model
• Lifetime statistics in chaotic dielectric microresonators
• A Note on Mixed Matrix Moments for the Complex Ginibre Ensemble
• Resonance distribution in open quantum chaotic systems
• Effective Hamiltonian and unitarity of the S matrix
• On matrix model partition functions for QCD with chemical potential
• Resonance width distribution in RMT: Weak coupling regime beyond Porter-Thomas
• Universality of spectra for interacting quantum chaotic systems
• Unquenched complex Dirac spectra at nonzero chemical potential: two-colour QCD lattice data versus matrix model
• The Interpolating Airy Kernels for the beta=1 and beta=4 Elliptic Ginibre Ensembles
• The chiral Gaussian two-matrix ensemble of real asymmetric matrices
• On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values
• A Wigner Surmise for Hermitian and Non-Hermitian Chiral Random Matrices
• Crossover from Conserving to Lossy Transport in Circular Random Matrix Ensembles
• Pseudo-Hermitian random matrix theory
• Quasimodes of a chaotic elastic cavity with increasing local losses
• Individual complex Dirac eigenvalue distributions from random matrix theory and comparison to quenched lattice QCD with a quark chemical potential
• Field theory of non-Hermitian disordered systems
• Scrambling and operator entanglement in local non-Hermitian quantum systems
• Chaotic Scattering with Localized Losses: S-Matrix Zeros and Reflection Time Difference for Systems with Broken Time Reversal Invariance
• Universality of the number variance in rotational invariant two-dimensional Coulomb gases
• Mode statistics in random lasers
• One-component plasma on a spherical annulus and a random matrix ensemble
• Closed-Form Density of States and Localization Length for a Non-Hermitian Disordered System
• Toward a classification of PT-symmetric quantum systems: From dissipative dynamics to topology and wormholes
• Signatures of Classical Diffusion in Quantum Fluctuations of 2D Chaotic Systems
• Matrix models and eigenvalue statistics for truncations of classical ensembles of random unitary matrices
• Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities
• Two-Dimensional Elliptic Determinantal Point Processes and Related Systems
• Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality
• Structure of trajectories of complex matrix eigenvalues in the Hermitian-non-Hermitian transition
• On the Largest Singular Values of Random Matrices with Independent Cauchy Entries
• Effective Non-Hermitian Hamiltonians for Studying Resonance Statistics in Open Disordered Systems
• On the Correlation Functions of the Characteristic Polynomials of Non-Hermitian Random Matrices with Independent Entries
• Random-matrix theory of amplifying and absorbing resonators with PT or PTT' symmetry
• Universal eigenvector correlations in quaternionic Ginibre ensembles
• Universal K-matrix distribution in beta=2 Ensembles of Random Matrices
• Hurwitz numbers from Feynman diagrams
• Extreme eigenvalues and the emerging outlier in rank-one non-Hermitian deformations of the Gaussian Unitary Ensemble
• Clustering of periodic orbits and ensembles of truncated unitary matrices
• Semiclassical theory of chaotic quantum resonances
• Dynamics of a rank-one perturbation of a Hermitian matrix
• Level Repulsion in Constrained Gaussian Random-Matrix Ensembles
• Nonhermitian Supersymmetric Partition Functions: the case of one bosonic flavor
• Locality, Correlations, Information, and non-Hermitian Quantum Systems
• Random matrices with prescribed eigenvalues and expectation values for random quantum states
• Universal spectral correlations in ensembles of random normal matrices
• Chaotic scattering through coupled cavities
• Pfaffian structure of the eigenvector overlap for the symplectic Ginibre ensemble
• On the Spectrum of Random Anti-symmetric and Tournament Matrices
• A path integral approach to sparse non-Hermitian random matrices
• Limiting eigenvalue distribution of the general deformed Ginibre ensemble
• Replica Symmetry Breaking in Random Non-Hermitian Systems
• Dynamics of a rank-one multiplicative perturbation of a unitary matrix
• A matrix model of a non-Hermitian $\beta$-ensemble
• Multiplicative non-Hermitian perturbations of classical $\beta$-ensembles