Abstract
The generalised eigenvalues for a pair of N × N matrices (X 1, X 2) are defined as the solutions of the equation det (X 1 − λX 2) = 0, or equivalently, for X 2 invertible, as the eigenvalues of \({X_{2}^{-1}X_{1}}\). We consider Gaussian real matrices X 1, X 2, for which the generalised eigenvalues have the rotational invariance of the half-sphere, or after a fractional linear transformation, the rotational invariance of the unit disk. In these latter variables we calculate the joint eigenvalue probability density function, the probability p N,k of finding k real eigenvalues, the densities of real and complex eigenvalues (the latter being related to an average over characteristic polynomials), and give an explicit Pfaffian formula for the higher correlation functions \({\rho_{(k_1,k_2)}}\). A limit theorem for p N,k is proved, and the scaled form of \({\rho_{(k_1,k_2)}}\) is shown to be identical to the analogous limit for the correlations of the eigenvalues of real Gaussian matrices. We show that these correlations satisfy sum rules characteristic of the underlying two-component Coulomb gas.
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Akemann G., Basile F.: Massive partition functions and complex eigenvalue correlations in matrix models with symplectic symmetry. Nucl. Phys. B 766, 150–177 (2007)
Akemann G., Kanzieper E.: Integrable structure of Ginibre’s ensemble of real random matrices and a Pfaffian integration theorem. J. Stat. Phys. 129, 1159–1231 (2007)
Akemann G., Phillips M.J., Sommers H.-J.: Characteristic polynomials in real Ginibre ensembles. J. Phys. A: Math. Theor. 42(1), 012001 (2008)
Akemann G., Vernizzi G.: Characteristic polynomials of complex matrix models. Nucl. Phys. B 660, 532–556 (2003)
Alastuey A., Jancovici B.: On the two-dimensional one-component Coulomb plasma. J. Physique 42, 1–12 (1981)
Bai Z.D.: Circular law. Ann. Probab. 25(1), 494–529 (1997)
Bender E.A.: Central and local limit theorems applied to asymptotic enumeration. J. Combin. Theory 15(1), 91–111 (1973)
Bordenave, C.: On the spectrum of sum and products of non-Hermitian random matrices. arXiv:1010.3087 (2010)
Borodin A., Sinclair C.D.: The Ginibre ensemble of real random matrices and its scaling limits. Commun. Math. Phys. 291, 177–224 (2009)
Caillol J.M.: Exact results for a two-dimensional one-component plasma on a sphere. Journal de Physique Lettres 42, L245 (1981)
Edelman A.: The probability that a random real Gaussian matrix has k real eigenvalues, related distributions, and the circular law. J. Multiv. Anal. 60, 203–232 (1997)
Edelman A., Kostlan E.: How many zeros of a random polynomial are real?. Am. Math. Soc. 32(1), 1–37 (1995)
Edelman A., Kostlan E., Shub M.: How many eigenvalues of a random matrix are real?. J. Am. Math. Soc. 7, 247–267 (1994)
Forrester P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010)
Forrester P.J.: The two-dimensional one-component plasma at Γ = 2: metallic boundary. J. Phys. A 18, 1419–1434 (1985)
Forrester P.J., Krishnapur M.: Derivation of an eigenvalue probability density function relating to the Poincaré disk. J. Phys. A: Math. Theor. 42, 385203 (2009)
Forrester P.J., Nagao T.: Eigenvalue statistics of the real Ginibre ensemble. Phys. Rev. Lett. 99, 050603 (2007)
Forrester P.J., Nagao T.: Skew-orthogonal polynomials and the partly symmetric real Ginibre ensemble. J. Phys. A: Math. Theor. 41, 375003 (2008)
Fyodorov Y.V., Khoruzhenko B.A.: On absolute moments of characteristic polynomials of a certain class of complex random matrices. Commun. Math. Phys. 273(3), 561–599 (2007)
Ginibre J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440–449 (1965)
Girko V.L.: Circular Law. Theory Probab. Appl. 29, 694–706 (1984)
Gradsteyn I.S., Ryzhik I.M.: Tables of Integrals, Series and Products. Academic Press, New York (1994)
Hough J.B., Krishnapur M., Peres Y., Virag B.: Determinantal processes and independence. Probab. Surv. 3, 206–229 (2006)
Jancovici B.: Classical Coulomb systems near a plane wall, II. J. Stat. Phys. 29, 263–280 (1982)
Kanzieper E.: Eigenvalue correlations in non-Hermitian symplectic random matrices. J. Phys. A: Math. Gen. 35, 6631–6644 (2002)
Krishnapur M.: From random matrices to random analytic functions. Ann. Probab. 37(1), 314–346 (2008)
Kolda T.G., Bader B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)
Kruskal J.B.: Rank, decomposition, and uniqueness for 3-way and N-way arrays. In: Coppi, R., Bolasco, S. (eds) Multiway Data Analysis, pp. 7–18. North-Holland, Amsterdam (1989)
Martin, C.D.: The rank of a 2 × 2 × 2 tensor. http://www.math.jmu.edu/~carlam/talks/Rank.pdf (2007)
Martin Ph.A.: Sum rules in charged fluids. Rev. Mod. Phys. 60, 1075–1127 (1988)
MacDonald B.: Density of complex zeros of a system of real random polynomials. J. Stat. Phys. 136, 807–833 (2009)
Mehta M.L.: Random Matrices and the Statistical Theory of Energy Levels. Academic Press, New York (1967)
Muirhead R.J.: Aspects of Multivariate Statistical Theory. Wiley, Hoboken (1982)
Sinclair, C.D.: Averages over Ginibre’s ensemble of random real matrices. Int. Math. Res. Not. 2007, rnm015 (2007)
Sinclair C.D.: Correlation functions for β = 1 ensembles of matrices of odd size. J. Stat. Phys. 136(1), 17–33 (2008)
Sommers H.-J., Wieczorek W.: General eigenvalue correlations for the real Ginibre ensemble. J. Phys. A: Math. Theor. 41, 405003 (2008)
Tao T., Vu V., Krishnapur M.: Random matrices: universality of ESDS and the circular law. Ann. Probab. 38(5), 2023–2065 (2008)
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Forrester, P.J., Mays, A. Pfaffian point process for the Gaussian real generalised eigenvalue problem. Probab. Theory Relat. Fields 154, 1–47 (2012). https://doi.org/10.1007/s00440-011-0361-8
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DOI: https://doi.org/10.1007/s00440-011-0361-8