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Pfaffian point process for the Gaussian real generalised eigenvalue problem
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  • Published: 11 April 2011

Pfaffian point process for the Gaussian real generalised eigenvalue problem

  • Peter J. Forrester1 &
  • Anthony Mays1 

Probability Theory and Related Fields volume 154, pages 1–47 (2012)Cite this article

  • 321 Accesses

  • 18 Citations

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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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References

  1. Akemann G., Basile F.: Massive partition functions and complex eigenvalue correlations in matrix models with symplectic symmetry. Nucl. Phys. B 766, 150–177 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Akemann G., Phillips M.J., Sommers H.-J.: Characteristic polynomials in real Ginibre ensembles. J. Phys. A: Math. Theor. 42(1), 012001 (2008)

    Article  MathSciNet  Google Scholar 

  4. Akemann G., Vernizzi G.: Characteristic polynomials of complex matrix models. Nucl. Phys. B 660, 532–556 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alastuey A., Jancovici B.: On the two-dimensional one-component Coulomb plasma. J. Physique 42, 1–12 (1981)

    MathSciNet  Google Scholar 

  6. Bai Z.D.: Circular law. Ann. Probab. 25(1), 494–529 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bender E.A.: Central and local limit theorems applied to asymptotic enumeration. J. Combin. Theory 15(1), 91–111 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bordenave, C.: On the spectrum of sum and products of non-Hermitian random matrices. arXiv:1010.3087 (2010)

  9. Borodin A., Sinclair C.D.: The Ginibre ensemble of real random matrices and its scaling limits. Commun. Math. Phys. 291, 177–224 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Caillol J.M.: Exact results for a two-dimensional one-component plasma on a sphere. Journal de Physique Lettres 42, L245 (1981)

    Article  Google Scholar 

  11. 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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Edelman A., Kostlan E.: How many zeros of a random polynomial are real?. Am. Math. Soc. 32(1), 1–37 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Edelman A., Kostlan E., Shub M.: How many eigenvalues of a random matrix are real?. J. Am. Math. Soc. 7, 247–267 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  14. Forrester P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010)

    MATH  Google Scholar 

  15. Forrester P.J.: The two-dimensional one-component plasma at Γ = 2: metallic boundary. J. Phys. A 18, 1419–1434 (1985)

    Article  MathSciNet  Google Scholar 

  16. 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)

    Article  MathSciNet  Google Scholar 

  17. Forrester P.J., Nagao T.: Eigenvalue statistics of the real Ginibre ensemble. Phys. Rev. Lett. 99, 050603 (2007)

    Article  Google Scholar 

  18. Forrester P.J., Nagao T.: Skew-orthogonal polynomials and the partly symmetric real Ginibre ensemble. J. Phys. A: Math. Theor. 41, 375003 (2008)

    Article  MathSciNet  Google Scholar 

  19. 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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ginibre J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440–449 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  21. Girko V.L.: Circular Law. Theory Probab. Appl. 29, 694–706 (1984)

    Article  MathSciNet  Google Scholar 

  22. Gradsteyn I.S., Ryzhik I.M.: Tables of Integrals, Series and Products. Academic Press, New York (1994)

    Google Scholar 

  23. Hough J.B., Krishnapur M., Peres Y., Virag B.: Determinantal processes and independence. Probab. Surv. 3, 206–229 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jancovici B.: Classical Coulomb systems near a plane wall, II. J. Stat. Phys. 29, 263–280 (1982)

    Article  MathSciNet  Google Scholar 

  25. Kanzieper E.: Eigenvalue correlations in non-Hermitian symplectic random matrices. J. Phys. A: Math. Gen. 35, 6631–6644 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Krishnapur M.: From random matrices to random analytic functions. Ann. Probab. 37(1), 314–346 (2008)

    Article  MathSciNet  Google Scholar 

  27. Kolda T.G., Bader B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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)

    Google Scholar 

  29. Martin, C.D.: The rank of a 2 × 2 × 2 tensor. http://www.math.jmu.edu/~carlam/talks/Rank.pdf (2007)

  30. Martin Ph.A.: Sum rules in charged fluids. Rev. Mod. Phys. 60, 1075–1127 (1988)

    Article  Google Scholar 

  31. MacDonald B.: Density of complex zeros of a system of real random polynomials. J. Stat. Phys. 136, 807–833 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mehta M.L.: Random Matrices and the Statistical Theory of Energy Levels. Academic Press, New York (1967)

    MATH  Google Scholar 

  33. Muirhead R.J.: Aspects of Multivariate Statistical Theory. Wiley, Hoboken (1982)

    Book  MATH  Google Scholar 

  34. Sinclair, C.D.: Averages over Ginibre’s ensemble of random real matrices. Int. Math. Res. Not. 2007, rnm015 (2007)

  35. Sinclair C.D.: Correlation functions for β = 1 ensembles of matrices of odd size. J. Stat. Phys. 136(1), 17–33 (2008)

    Article  MathSciNet  Google Scholar 

  36. Sommers H.-J., Wieczorek W.: General eigenvalue correlations for the real Ginibre ensemble. J. Phys. A: Math. Theor. 41, 405003 (2008)

    Article  MathSciNet  Google Scholar 

  37. Tao T., Vu V., Krishnapur M.: Random matrices: universality of ESDS and the circular law. Ann. Probab. 38(5), 2023–2065 (2008)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Melbourne, Victoria, 3010, Australia

    Peter J. Forrester & Anthony Mays

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  1. Peter J. Forrester
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  2. Anthony Mays
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Correspondence to Anthony Mays.

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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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  • Received: 01 April 2010

  • Revised: 18 February 2011

  • Published: 11 April 2011

  • Issue Date: October 2012

  • DOI: https://doi.org/10.1007/s00440-011-0361-8

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Mathematics Subject Classification (2010)

  • 60B20 (15B52, 82B41)
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