Abstract
In the recent publication (E. Kanzieper and G. Akemann in Phys. Rev. Lett. 95:230201, 2005), an exact solution was reported for the probability p n,k to find exactly k real eigenvalues in the spectrum of an n×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.
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Adler, M., Forrester, P.J., Nagao, T., van Moerbeke, P.: Classical skew orthogonal polynomials and random matrices. J. Stat. Phys. 99, 141 (2000)
Agam, O., Bettelheim, E., Wiegmann, P.B., Zabrodin, A.: Viscous fingering and the shape of an electronic droplet in the quantum Hall regime. Phys. Rev. Lett. 88, 236801 (2002)
Akemann, G.: The complex Laguerre symplectic ensemble of non-Hermitean matrices. Nucl. Phys. B 730, 253 (2005)
Akemann, G.: Matrix models and QCD with chemical potential. Int. J. Mod. Phys. A 22, 1077 (2007)
Akemann, G., Basile, F.: Massive partition functions and complex eigenvalue correlations in matrix models with symplectic symmetry. Nucl. Phys. B 766, 150 (2007)
Andrews, G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998)
Bai, Z.D.: Circular law. Ann. Probab. 25, 494 (1997)
Borodin, A., Sinclair, C.D.: Correlation functions of ensembles of asymmetric real matrices. arXiv: 0706.2670 (2007)
Borodin, A., Strahov, E.: Averages of characteristic polynomials in random matrix theory. Commun. Pure Appl. Math. LVIII, 0001 (2005)
Chalker, J.T., Mehlig, B.: Eigenvector statistics in non-Hermitean random matrix ensembles. Phys. Rev. Lett. 81, 3367 (1998)
Dyson, F.J.: Correlations between eigenvalues of a random matrix. Commun. Math. Phys. 19, 235 (1970)
Dyson, F.J.: Quaternion determinants. Helv. Phys. Acta 49, 289 (1972)
Edelman, A.: The probability that a random real Gaussian matrix has k real eigenvalues. Related distributions, and the circular law. J. Mult. Anal. 60, 203 (1997)
Edelman, A., Kostlan, E., Shub, M.: How many eigenvalues of a random matrix are real? J. Am. Math. Soc. 7, 247 (1994)
Efetov, K.B.: Directed quantum chaos. Phys. Rev. Lett. 79, 491 (1997)
Efetov, K.B.: Quantum disordered systems with a direction. Phys. Rev. B 56, 9630 (1997)
Eynard, B.: Asymptotics of skew orthogonal polynomials. J. Phys. A: Math. Gen. 34, 7591 (2001)
Forrester, P.J. Log-Gases and Random Matrices. Web-book (2005)
Forrester, P.J., Nagao, T.: Eigenvalue statistics of the real Ginibre ensemble. Phys. Rev. Lett. 99, 050603 (2007)
Fyodorov, Y.V., Sommers, H.-J.: Random matrices close to Hermitean or unitary: Overview of methods and results. J. Phys. A: Math. Gen. 36, 3303 (2003)
Fyodorov, Y.V., Khoruzhenko, B., Sommers, H.-J.: Almost Hermitean random matrices: Crossover from Wigner-Dyson to Ginibre eigenvalue statistics. Phys. Rev. Lett. 79, 557 (1997)
Ginibre, J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 19, 133 (1965)
Girko, V.L.: Circle law. Theory Probab. Appl. 29, 694 (1984)
Girko, V.L.: Elliptic law. Theory Probab. Appl. 30, 677 (1986)
Grobe, R., Haake, F., Sommers, H.-J.: Quantum distinction of regular and chaotic dissipative motion. Phys. Rev. Lett. 61, 1899 (1988)
Grobe, R., Haake, F.: Universality of cubic-level repulsion for dissipative quantum chaos. Phys. Rev. Lett. 62, 2893 (1989)
Guhr, T., Müller-Groeling, A., Weidenmüller, H.A.: Random matrix theories in quantum physics: Common concepts. Phys. Reports 299, 189 (1998)
Halasz, M.A., Osborn, J.C., Verbaarschot, J.J.M.: Random matrix triality at nonzero chemical potential. Phys. Rev. D 56, 7059 (1997)
Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. Lond. Math. Soc. B 17, 75 (1918)
Jack, H.: A class of polynomials in search of a definition, or the symmetric group parameterized. In: Kuznetsov, V.B., Sahi, S. (eds.) Jack, Hall-Littlewood and Macdonald polynomials. Contemporary Mathematics Series. AMS, Providence (2006)
Janik, R.A., Nörenberg, W., Nowak, M.A., Papp, G., Zahed, I.: Correlations of eigenvectors for non-Hermitean random-matrix models. Phys. Rev. E 60, 2699 (1999)
Kanzieper, E.: Eigenvalue correlations in non-Hermitean symplectic random matrices. J. Phys. A: Math. Gen. 35, 6631 (2002)
Kanzieper, E.: Replica field theories, Painlevé transcendents, and exact correlation functions. Phys. Rev. Lett. 89, 250201 (2002)
Kanzieper, E.: Exact replica treatment of non-Hermitean complex random matrices. In: Kovras, O. (ed.) Frontiers in Field Theory, p. 23. Nova Science Publishers, New York (2005)
Kanzieper, E., Akemann, G.: Statistics of real eigenvalues in Ginibre’s ensemble of random real matrices. Phys. Rev. Lett. 95, 230201 (2005)
Khoruzhenko, B.A., Mezzadri, F.: Private communication (2005)
Kolesnikov, A.V., Efetov, K.B.: Distribution of complex eigenvalues for symplectic ensembles of non-Hermitean matrices. Waves Random Media 9, 71 (1999)
Kwapień, J., Drożdż, S., Ioannides, A.A.: Temporal correlations versus noise in the correlation matrix formalism: An example of the brain auditory response. Phys. Rev. E 62, 5557 (2000)
Kwapień, J., Drożdż, S., Górski, A.Z., Oświęcimka, P.: Asymmetric matrices in an analysis of financial correlations. Acta Phys. Polonica B37, 3039 (2006)
Le Caër, G., Ho, J.S.: The Voronoi tessellation generated from eigenvalues of complex random matrices. J. Phys. A: Math. Gen. 23, 3279 (1990)
Le Caër, G., Delannay, R.: Topological models of 2D fractal cellular structures. J. Phys. I (France) 3, 1777 (1993)
Lehmann, N., Sommers, H.J.: Eigenvalue statistics of random real matrices. Phys. Rev. Lett. 67, 941 (1991)
Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford University Press, Oxford (1998)
Mahoux, G., Mehta, M.L.: A method of integration over matrix variables. J. Phys. I (France) 1, 1093 (1991)
Markum, H., Pullirsch, R., Wettig, T.: Non-Hermitean random matrix theory and lattice QCD with chemical potential. Phys. Rev. Lett. 83, 484 (1999)
Mehlig, B., Chalker, J.T.: Statistical properties of eigenvectors in non-Hermitean Gaussian random matrix ensembles. J. Math. Phys. 41, 3233 (2000)
Mehta, M.L.: A note on certain multiple integrals. J. Math. Phys. 17, 2198 (1976)
Mehta, M.L.: Random Matrices. Elsevier, Amsterdam (2004)
Mehta, M.L., Srivastava, P.K.: Correlation functions for eigenvalues of real quaternion matrices. J. Math. Phys. 7, 341 (1966)
Mineev-Weinstein, M., Wiegmann, P.B., Zabrodin, A.: Integrable structure of interface dynamics. Phys. Rev. Lett. 84, 5106 (2000)
Muirhead, R.J.: Aspects of Multivariate Statistical Theory. Wiley, New York (1982)
Nagao, T., Nishigaki, S.M.: Massive random matrix ensembles at β=1 and 4: QCD in three dimensions. Phys. Rev. D 63, 045011 (2001)
Nishigaki, S.M., Kamenev, A.: Replica treatment of non-Hermitean disordered Hamiltonians. J. Phys. A: Math. Gen. 35, 4571 (2002)
Osborn, J.C.: Universal results from an alternate random matrix model for QCD with a baryon chemical potential. Phys. Rev. Lett. 93, 222001 (2004)
Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, vol. 2. Gordon and Breach, New York (1986)
Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, vol. 3. Gordon and Breach, New York (1990)
Sinclair, C.D.: Averages over Ginibre’s ensemble of random real matrices. Int. Math. Res. Not. 2007, rnm015 (2007)
Sommers, H.J.: Symplectic structure of the real Ginibre ensemble. J. Phys. A: Math. Theor. 40, F671 (2007)
Sommers, H.J., Crisanti, A., Sompolinsky, H., Stein, Y.: Spectrum of large random asymmetric matrices. Phys. Rev. Lett. 60, 1895 (1988)
Sompolinsky, H., Crisanti, A., Sommers, H.J.: Chaos in random neural networks. Phys. Rev. Lett. 61, 259 (1988)
Splittorff, K., Verbaarschot, J.J.M.: Factorization of correlation functions and the replica limit of the Toda lattice equation. Nucl. Phys. B 683, 467 (2004)
Stephanov, M.: Random matrix model of QCD at finite density and the nature of the quenched limit. Phys. Rev. Lett. 76, 4472 (1996)
Timme, M., Wolf, F., Geisel, T.: Coexistence of regular and irregular dynamics in complex networks of pulse-coupled oscillators. Phys. Rev. Lett. 89, 258701 (2002)
Timme, M., Wolf, F., Geisel, T.: Topological speed limits to network synchronization. Phys. Rev. Lett. 92, 074101 (2004)
Tracy, C.A., Widom, H.: Correlation functions, cluster functions, and spacing distributions for random matrices. J. Stat. Phys. 92, 809 (1998)
Wigner, E.P.: Statistical properties of real symmetric matrices with many dimensions. In: Proc. 4th Can. Math. Cong. (Toronto), p. 174 (1957)
Wigner, E.P.: The unreasonable effectiveness of mathematics in natural sciences. Commun. Pure Appl. Math. 13, 1 (1960)
Zabrodin, A.: New applications of non-Hermitean random matrices. In: Iagolnitzer, D., Rivasseau, V., Zinn-Justin, J. (eds.) Proceedings of the International Conference on Theoretical Physics (TH-2002). Birkhäuser, Basel (2003)
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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). https://doi.org/10.1007/s10955-007-9381-2
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DOI: https://doi.org/10.1007/s10955-007-9381-2