Integrable Structure of Ginibre's Ensemble of Real Random Matrices and a Pfaffian Integration Theorem

Abstract

In the recent publication [E. Kanzieper and G. Akemann, Phys. Rev. Lett. 95, 230201 (2005); arXiv: math-ph/0507058], an exact solution was reported for the probability "p_{n,k}" to find exactly "k" real eigenvalues in the spectrum of an "n" by "n" real asymmetric matrix drawn at random from Ginibre's Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof of the Pfaffian integration theorem, the key ingredient of our analysis of the statistics of real eigenvalues in the GinOE. We also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a GinOE matrix restricted to have exactly "k" real eigenvalues. In the particular case of "k=0", all correlation functions of complex eigenvalues are determined.

73 pages, 6 figures, 2 tables

References in corpus (4)

• Eigenvalue statistics of the real Ginibre ensemble
• Statistics of Real Eigenvalues in Ginibre's Ensemble of Random Real Matrices
• Matrix Models and QCD with Chemical Potential
• Massive partition functions and complex eigenvalue correlations in Matrix Models with symplectic symmetry

Cited by in corpus (39)

• Around the circular law
• The Ginibre ensemble of real random matrices and its scaling limits
• Random matrices: Universality of local spectral statistics of non-Hermitian matrices
• Spectrum of the Product of Independent Random Gaussian Matrices
• Nonlinear Analogue of the May-Wigner Instability Transition
• Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble
• Spectral Theory of Sparse Non-Hermitian Random Matrices
• Determinant formula for the octagon form factor in $\mathcal{N}$=4 SYM
• Circular Law Theorem for Random Markov Matrices
• Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation
• General Eigenvalue Correlations for the Real Ginibre Ensemble
• Exact persistence exponent for the $2d$-diffusion equation and related Kac polynomials
• Eigenvalue Density of the non-Hermitian Wilson Dirac Operator
• The Octagon as a Determinant
• Gap Probabilities in Non-Hermitian Random Matrix Theory
• Characteristic polynomials in real Ginibre ensembles
• A new approach to derive Pfaffian structures for random matrix ensembles
• Level statistics of a pseudo-Hermitian Dicke model
• Eigenvalue Separation in Some Random Matrix Models
• Bures-Hall Ensemble: Spectral Densities and Average Entropies
• A method to calculate correlation functions for $\beta =1$ random matrices of odd size
• Matrix averages relating to the Ginibre ensembles
• Eigenvalue Repulsion and Eigenfunction Localization in Sparse Non-Hermitian Random Matrices
• Asymmetric correlation matrices: an analysis of financial data
• Universality of spectra for interacting quantum chaotic systems
• The chiral Gaussian two-matrix ensemble of real asymmetric matrices
• Non-Hermitean Wishart random matrices (I)
• Spectral Domain of Large Nonsymmetric Correlated Wishart Matrices
• What is the probability that a large random matrix has no real eigenvalues?
• The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices
• Finite size corrections for real eigenvalues of the elliptic Ginibre matrices
• On the Correlation Functions of the Characteristic Polynomials of Non-Hermitian Random Matrices with Independent Entries
• Matrix Quantum Mechanics on $S^1/{\mathbb{Z}}_2$
• Real spectra of large real asymmetric random matrices
• Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge
• Random Lindblad operators obeying detailed balance
• The probability of almost all eigenvalues being real for the elliptic real Ginibre ensemble
• Spectral Statistics of Non-Hermitian Matrices and Dissipative Quantum Chaos
• Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach