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Diffuse Waves in Complex Media pp 165–211Cite as

Spectra of Large Random Matrices: A Method of Study

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Part of the book series: NATO Science Series ((ASIC,volume 531))

Abstract

A formalism for study of spectral correlations in non-Gaussian, unitary invariant ensembles of large random matrices with strong level confinement is reviewed. It is based on the Shohat method in the theory of orthogonal polynomials. The approach presented is equally suitable for description of both local and global spectral characteristics, thereby providing an overall look at the phenomenon of spectral universality in Random Matrix Theory.

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References

  1. Mehta, M. L. (1991) Random Matrices, Academic Press, Boston.

    MATH  Google Scholar 

  2. Guhr, T., Müller-Groeling, A., and Weidenmüller, H. A. (1998) Random matrix theories in quantum physics: Common concepts, Phys. Rep. 299, 189–428.

    Article  ADS  MathSciNet  Google Scholar 

  3. Wigner, E. P. (1951) On the statistical distribution of the widths and spacings of nuclear resonance levels, Proc. Cambridge Philos. Soc. 47, 790–798.

    Article  ADS  MATH  Google Scholar 

  4. Dyson, F. J. (1962) Statistical theory of energy levels of complex systems I, II, III, J. Math. Phys. 3, 140–156; 3, 157–165; 3, 166–175.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Dyson, F. J. (1962) The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics, J. Math. Phys. 3, 1199–1215.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Brody, T. A., Flores, J., French, J. B., Mello, P. A., Pandey, A., and Wong, S. S. M. (1981) Random-matrix physics: Spectrum and strength fluctuations, Rev. Mod. Phys. 53, 385–479.

    Article  ADS  MathSciNet  Google Scholar 

  7. Di Francesco, P., Ginsparg, P., and Zinn-Justin, J. (1995) 2D gravity and random matrices, Phys. Rep. 254, 1–133.

    Article  ADS  Google Scholar 

  8. Verbaarschot, J. J. M. and Zahed, I. (1993) Spectral density of the QCD Dirac operator near zero virtuality, Phys. Rev. Lett. 70, 3852–3855; Verbaarschot, J. J. M. (1994) The spectrum of the QCD Dirac operator and chiral random matrix theory: the threefold way, Phys. Rev. Lett. 72, 2531–2533; Verbaarschot, J. J. M. and Zahed, I. (1994) Random matrix theory and QCD, Phys. Rev. Lett. 73, 2288–2291.

    Article  ADS  Google Scholar 

  9. Gutzwiller, M. C. (1990) Chaos in Classical and Quantum Mechanics, Springer-Verlag, New York.

    MATH  Google Scholar 

  10. Altshuler, B. L., Lee, P. A., and Webb, R. A. (1991) Mesoscopic Phenomena in Solids, North-Holland, Amsterdam.

    Google Scholar 

  11. Beenakker, C. W. J. (1997) Random matrix theory of quantum transport, Rev. Mod. Phys. 69, 731–808.

    Article  ADS  Google Scholar 

  12. Gor’kov, L. P. and Eliashberg, G. M. (1965) Minute metallic particles in an electromagnetic field, Zh. Eksp. Teor. Fiz. 48, 1407–1418 [Sov. Phys. JETP 21, 940–947].

    Google Scholar 

  13. Porter, C. E. (1965) Statistical theories of spectra: Fluctuations, Academic Press, New York.

    Google Scholar 

  14. Efetov, K. B. (1983) Supersymmetry and theory of disordered metals, Adv. Phys. 32, 53–127.

    Article  ADS  MathSciNet  Google Scholar 

  15. Berbenni-Bitsch, M. E., Meyer, S., Schäfer, A., Verbaarschot, J. J. M., and Wettig, T. (1998) Microscopic universality in the spectrum of the lattice Dirac operator, Phys. Rev. Lett. 80, 1146–1149.

    Article  ADS  Google Scholar 

  16. Verbaarschot, J. J. M. (1998) Universal fluctuations in Dirac spectra, in New Developments in Quantum Field Theory, Plenum Press, New York.

    Google Scholar 

  17. Akemann, G., Damgaard, P. H., Magnea, U., and Nishigaki, S. (1997) Universality of random matrices in the microscopic limit and the Dirac operator spectrum, Nucl. Phys. B 487, 721–738.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Dyson, F. J. (1972) A class of matrix ensembles, J. Math. Phys. 13, 90–97.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. Ambjørn, J., Jurkiewicz, J., and Makeenko, Yu. M. (1990) Multiloop correlators for two-dimensional quantum gravity, Phys. Lett. B 251, 517–524.

    Article  ADS  MathSciNet  Google Scholar 

  20. Akemann, G. and Ambjørn, J. (1996) New universal spectral correlators, J. Phys. A 29, L555–L560.

    Article  ADS  MATH  Google Scholar 

  21. Itoi, C. (1997) Universal wide correlators in non-Gaussian orthogonal, unitary and symplectic random matrix ensembles, Nucl. Phys. B 493, 651–659.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. Beenakker, C. W. J. (1993) Universality in the random-matrix theory of quantum transport, Phys. Rev. Lett. 70, 1155–1158.

    Article  ADS  Google Scholar 

  23. Beenakker, C. W. J. (1994) Universality of Brézin and Zee’s spectral correlator, Nucl. Phys. B 422, 515–520.

    Article  ADS  Google Scholar 

  24. Brézin, E. and Zee, A. (1994) Correlation functions in disordered systems, Phys. Rev. E 49, 2588–2596.

    Article  ADS  Google Scholar 

  25. Hackenbroich, G. and Weidenmüller, H. A. (1995) Universality of random-matrix results for non-Gaussian ensembles, Phys. Rev. Lett. 74, 4118–4121.

    Article  ADS  Google Scholar 

  26. Verbaarschot, J. J. M., Weidenmüller, H. A., and Zirnbauer, M. R. (1985) Grass-mann integration in stochastic quantum physics: The case of compound-nucleus scattering, Phys. Rep. 129, 367–438.

    Article  ADS  MathSciNet  Google Scholar 

  27. Mehta, M. L. (1960) On the statistical properties of the level-spacings in nuclear spectra, Nucl. Phys. 18, 395–419; Mehta, M. L. and Gaudin, M. (1960) On the density of eigenvalues of a random matrix, Nucl. Phys. 18, 420–427.

    Article  MATH  Google Scholar 

  28. Fox, D. and Kahn, P. B. (1964) Higher order spacing distributions for a class of unitary ensembles, Phys. Rev. 134, B1151–B1155.

    Article  ADS  MathSciNet  Google Scholar 

  29. Leff, H. S. (1964) Class of ensembles in the statistical theory of energy-level spectra, J. Math. Phys. 5, 763–768.

    Article  ADS  MathSciNet  Google Scholar 

  30. Bronk, B. V. (1965) Exponential ensemble for random matrices, J. Math. Phys. 6, 228–237.

    Article  ADS  Google Scholar 

  31. Nagao, T. and Wadati, M. (1991) Correlation functions of random matrix ensembles related to classical orthogonal polynomials I, J. Phys. Soc. Jpn. 60, 3298–3322; Nagao, T. and Wadati, M. (1992) Correlation functions of random matrix ensembles related to classical orthogonal polynomials II, III, J. Phys. Soc. Jpn. 61, 78–88; 61, 1910–1918.

    Article  ADS  MathSciNet  Google Scholar 

  32. Basor, E. L., Tracy, C. A., and Widom, H. (1992) Asymptotics of level spacing distribution functions for random matrices, Phys. Rev. Lett. 69, 5–8.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  33. Brézin, E. and Neuberger, H. (1991) Multicritical points of unoriented random surfaces, Nucl. Phys. B 350, 513–553.

    Article  ADS  Google Scholar 

  34. Tracy, C. A. and Widom, H. (1998) Correlations functions, cluster functions and spacing distributions for random matrices, Los Alamos preprint archive, solv-int/9804004.

    Google Scholar 

  35. Mahoux, G. and Mehta, M. L. (1991) A method of integration over matrix variables: IV, J. Phys. (France) 1, 1093–1108.

    Article  MathSciNet  Google Scholar 

  36. Brézin, E. and Zee, A. (1993) Universality of the correlations between eigenvalues of large random matrices, Nucl. Phys. B 402, 613–627.

    Article  ADS  MATH  Google Scholar 

  37. Freilikher, V., Kanzieper, E., and Yurkevich, I. (1996) Unitary random-matrix ensemble with governable level confinement, Phys. Rev. E 53, 2200–2209.

    Article  ADS  MathSciNet  Google Scholar 

  38. Freilikher, V., Kanzieper, E., and Yurkevich, I. (1996) Theory of random matrices with strong level confinement: Orthogonal polynomial approach, Phys. Rev. E 54, 210–219.

    Article  ADS  MathSciNet  Google Scholar 

  39. Shohat, J. A. and Tamarkin, J. D. (1963) The problem of moments, American Mathematical Society, Providence.

    Google Scholar 

  40. Szegö, G. (1921) Über die Randwerte analytischer Funktionen, Math. Annalen 84, 232–244.

    Article  MATH  Google Scholar 

  41. Slevin, K. and Nagao, T. (1993) New random matrix theory of scattering in meso-scopic systems, Phys. Rev. Lett. 70, 635–638.

    Article  ADS  Google Scholar 

  42. Altland, A. and Zirnbauer, M. R. (1997) Nonstandard symmetry classes in meso-scopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142–1161.

    Article  ADS  Google Scholar 

  43. Nagao, T. and Slevin, K. (1993) Nonuniversal correlations for random matrix ensembles, J. Math. Phys. 34, 2075–2085; Laguerre ensembles of random matrices: Nonuniversal correlation functions, J. Math. Phys. 34, 2317–2330.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  44. Forrester, P. J. (1993) The spectrum of random matrix ensembles, Nucl. Phys. B 402, 709–728.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  45. Nagao, T. and Forrester, P. J. (1995) Asymptotic correlations at the spectrum edge of random matrices, Nucl. Phys. B 435, 401–420.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  46. Tracy, C. A. and Widom, H. (1994) Level-spacing distributions and the Bessel kernel, Commun. Math. Phys. 161, 289–309.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  47. Nishigaki, S. (1996) Proof of universality of the Bessel kernel for chiral matrix models in the microscopic limit, Phys. Lett. B 387, 139–144.

    Article  ADS  MathSciNet  Google Scholar 

  48. Kanzieper, E. and Freilikher, V. (1998) Random-matrix models with the logarithmic-singular level confinement: Method of fictitious fermions, Philos. Mag. B 77, 1161–1171.

    Article  ADS  Google Scholar 

  49. Wigner, E. P. (1962) Distribution laws for the roots of a random Hermitean matrix, reprinted in: Porter, C. E. (1965) Statistical theories of spectra: Fluctuations, pp. 446–461, Academic Press, New York.

    Google Scholar 

  50. Bowick, M. J. and Brézin, E. (1991) Universal scaling of the tail of the density of eigenvalues in random matrix models, Phys. Lett. B 268, 21–28.

    Article  ADS  MathSciNet  Google Scholar 

  51. Kanzieper, E. and Freilikher, V. (1997) Universality in invariant random-matrix models: Existence near the soft edge, Phys. Rev. E 55, 3712–3715.

    Article  ADS  Google Scholar 

  52. Tracy, C. A. and Widom, H. (1994) Level-spacing distributions and the Airy kernel, Commun. Math. Phys. 159, 151–174.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  53. Kanzieper, E. and Freilikher, V. (1997) Novel universal correlations in invariant random-matrix models, Phys. Rev. Lett. 78, 3806–3809.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  54. Damgaard, P. H. and Nishigaki, S. M. (1998) Universal spectral correlators and massive Dirac operators, Nucl. Phys. B 518, 495–512; Universal massive spectral correlators and QCD3, Phys. Rev. D 57, 5299–5302.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  55. Akemann, G. (1996) Higher genus correlators for the Hermitian matrix model with multiple cuts, Nucl. Phys. B 482, 403–430.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  56. Kanzieper, E. and Freilikher, V. (1998) Two-band random matrices, Phys. Rev. E 57, 6604–6611.

    Article  ADS  Google Scholar 

  57. Shohat, J. (1930) C. R. Hebd. Seances Acad. Sci. 191, 989; Shohat, J. (1939) A differential equation for orthogonal polynomials, Duke Math. J. 5, 401–417.

    Google Scholar 

  58. Sener, M. K. and Verbaarschot, J. J. M. (1998) Universality in chiral random matrix theory at β = 1 and β = 4, Los Alamos preprint archive, hep-th/9801042.

    Google Scholar 

  59. Widom, H. (1998) On the relation between orthogonal, symplectic and unitary matrix ensembles, Los Alamos preprint archive, solv-int/9804005.

    Google Scholar 

  60. Szegö, G. (1967) Orthogonal Polynomials, American Mathematical Society, Providence.

    Google Scholar 

  61. Bonan, S. S. and Clark, D. S. (1986) Estimates of the orthogonal polynomials with weight exp(−x m), m an even positive integer, J. Appr. Theory 46, 408–410.

    Article  MATH  MathSciNet  Google Scholar 

  62. Sheen, R. C. (1987) Plancherel-Rotach type asymptotics for orthogonal polynomials associated with exp (-x 6/6), J. Appr. Theory 30, 232–293; Bonan, S. S. and Clark, D. S. (1990) Estimates of the Hermite and the Freud polynomials, J. Appr. Theory 63, 210–224; Bauldry, W. C. (1990) Estimates of asymmetric Freud polynomials, J. Appr. Theory 63, 225–237; Mhaskar, H. N. (1990) Bounds for certain Freud type orthogonal polynomials, J. Appr. Theory 63, 238–254.

    Article  MathSciNet  Google Scholar 

  63. Nevai, P. (1986) Géza Freud, orthogonal polynomials and Christoffel functions: A case study, J. Appr. Theory 48, 3–167.

    Article  MATH  Google Scholar 

  64. Lubinsky, D. S. (1993) An update on orthogonal polynomials and weighted approximation on the real line, Acta Appl. Math. 33, 121–164.

    Article  MATH  MathSciNet  Google Scholar 

  65. Akemann, G. and Kanzieper, E. (1998) unpublished.

    Google Scholar 

  66. Akemann, G., Damgaard, P. H., Magnea, U., and Nishigaki, S. (1998) Multicritical microscopic spectral correlators of Hermitian and complex matrices, Nucl. Phys. B 519, 682–714.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  67. Gross, D. J. and Migdal, A. A. (1990) Nonperturbative two-dimensional quantum gravity, Phys. Rev. Lett. 64, 127–130.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  68. Dingle, R. B. (1973) Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, New York.

    MATH  Google Scholar 

  69. Molinari, L. (1988) Phase structure of matrix models through orthogonal polynomials, J. Phys. A 21, 1–6.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  70. Douglas, M., Seiberg, N., and Shenker, S. (1990) Flow and instability in quantum gravity, Phys. Lett. B 244, 381–386.

    Article  ADS  MathSciNet  Google Scholar 

  71. Sasaki, M. and Suzuki, H. (1991) Matrix realization of random surfaces, Phys. Rev. D 43, 4015–4028.

    Article  ADS  MathSciNet  Google Scholar 

  72. Jurkiewicz, J., Nowak, M. A., and Zahed, I. (1996) Dirac spectrum in QCD and quark masses, Nucl. Phys. B 478, 605–626.

    Article  ADS  Google Scholar 

  73. Jackson, A. and Verbaarschot, J. (1996) Random matrix model for chiral symmetry breaking, Phys. Rev. D 53, 7223–7230.

    Article  ADS  Google Scholar 

  74. Cugliandolo, L. F., Kurchan, J., Parisi, G., and Ritort, F. (1995) Matrix models as solvable glass models, Phys. Rev. Lett. 74, 1012–1015.

    Article  ADS  Google Scholar 

  75. Morita, Y., Hatsugai, Y., and Kohmoto, M. (1995) Universal correlations in random matrices and one-dimensional particles with long-range interactions in a confinement potential, Phys. Rev. B 52, 4716–4719. See also Ref. [80].

    Article  ADS  Google Scholar 

  76. Shimamune, Y. (1982) On the phase structure of large N matrix models and gauge models, Phys. Lett. B 108, 407–410.

    Article  ADS  Google Scholar 

  77. Cicuta, G. M., Molinari, L., and Montaldi, E. (1990) Multicritical points in matrix models, J. Phys. A 23, L421–L425.

    Article  ADS  Google Scholar 

  78. Jurkiewicz, J. (1991) Chaotic behaviour in one-matrix models, Phys. Lett. B 261, 260–268.

    Article  ADS  MathSciNet  Google Scholar 

  79. Demeterfi, K., Deo, N., Jain, S., and Tan, C.-I. (1990) Multiband structure and critical behavior of matrix models, Phys. Rev. D 42, 4105–4122.

    Article  ADS  Google Scholar 

  80. Higuchi, S., Itoi, C., Nishigaki, S. M., and Sakai, N. (1997) Renormalization group approach to multiple-arc random matrix models, Phys. Lett. B 398, 123–129.

    Article  ADS  MathSciNet  Google Scholar 

  81. Deo, N. (1997) Orthogonal polynomials and exact correlation functions for two cut random matrix models, Nucl. Phys. B 504, 609–620.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  82. Altshuler, B. L. and Shklovskii, B. I. (1986) Repulsion of energy levels and conductivity of small metal samples, Zh. Eksp. Teor. Fiz. 91, 220–234 [Sov. Phys. JETP 64, 127–135].

    Google Scholar 

  83. Jalabert, R. A., Pichard, J.-L., and Beenakker, C. W. J. (1993) Long-range energy level interaction in small metallic particles, Europhys. Lett. 24, 1–6.

    Article  ADS  Google Scholar 

  84. Brézin, E. and Deo, N. (1998) Smoothed correlators for symmetric double-well matrix models: Some puzzles and resolutions, Los Alamos preprint archive, cond-mat/9805096.

    Google Scholar 

  85. Brézin, E. and Zinn-Justin, J. (1992) Renormalization group approach to matrix models, Phys. Lett. B 288, 54–58.

    Article  ADS  MathSciNet  Google Scholar 

  86. Muttalib, K. A., Chen, Y., Ismail, M. E. H., and Nicopoulos, V. N. (1993) New family of unitary random matrices, Phys. Rev. Lett. 71, 471–475.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  87. Bogomolny, E., Bohigas, O., and Pato, M. P. (1997) Distribution of eigenvalues of certain matrix ensembles, Phys. Rev. E 55, 6707–6718.

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. The Abdus Salam International Centre for Theoretical Physics, P.O.B. 586, 34100, Trieste, Italy

    E. Kanzieper

  2. The Jack and Pearl Resnick Institute for Advanced Technology Department of Physics, Bar-Ilan University, 52900, Ramat-Gan, Israel

    V. Freilikher

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  1. E. Kanzieper
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  2. V. Freilikher
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Editors and Affiliations

  1. CNRS, Ecole Polytechnique, France

    Jean-Pierre Fouque

  2. Department of Mathematics, North Carolina State University, Raleigh, NC, USA

    Jean-Pierre Fouque

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Kanzieper, E., Freilikher, V. (1999). Spectra of Large Random Matrices: A Method of Study. In: Fouque, JP. (eds) Diffuse Waves in Complex Media. NATO Science Series, vol 531. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4572-5_7

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