Communications in Mathematical Physics

, Volume 273, Issue 3, pp 561–599 | Cite as

On Absolute Moments of Characteristic Polynomials of a Certain Class of Complex Random Matrices

  • Yan. V. Fyodorov
  • Boris. A. KhoruzhenkoEmail author


The integer moments of the spectral determinant | det (zI − W) |2 of complex random matrices W are obtained in terms of the characteristic polynomial of the positive-semidefinite matrix WW for the class of matrices W = AU, where A is a given matrix and U is random unitary. This work is motivated by studies of complex eigenvalues of random matrices and potential applications of the obtained results are discussed in this context.


Random Matrice Random Matrix Characteristic Polynomial Random Matrix Theory Eigenvalue Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akemann G. and Vernizzi G. (2003). Characteristic Polynomials of Complex Random Matrix Models. Nucl. Phys. B 660: 532–556 zbMATHMathSciNetADSGoogle Scholar
  2. 2.
    Akemann G. and Pottier A. (2004). Ratios of characteristic polynomials in complex matrix models. J. Phys. A: Math and General 37: L453–L460 zbMATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Andreev A.V. and Simons B.D. (1995). Correlators of spectral determinants in quantum chaos. Phys. Rev. Lett. 75: 2304–2307 CrossRefADSGoogle Scholar
  4. 4.
    Balantekin, A.B.: Character expansions, Itzykson-Zuber integrals, and the QCD partition function. Phys. Rev. D(3) 62, 085017–085023 (2000)Google Scholar
  5. 5.
    Baik J., Deift P. and Strahov E. (2003). Products and ratios of characteristic polynomials of random Hermitian matrices. J. Math. Phys. 44: 3657–3670 zbMATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Berezin, F.A.: Some remarks on the Wigner distribution (in Russian). Teor. Mat. Fiz. 17, 305–318 (1973). English translation: Theoret. and Math. Phys. 17(3), 1163–1171 (1974)Google Scholar
  7. 7.
    Berezin, F.A.: Quantization in complex symmetric spaces (in Russian). Izv Akad Nauk SSSR, Ser Math 39, 363–402 (1975); English translation: Math USSR-Izv 9(2), 341–379 (1976)Google Scholar
  8. 8.
    Biane Ph. and Lehner F. (2001). Computation of some examples of Brown’s spectral measure in free probability. Colloq. Math. 90: 181–211 zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Borodin A., Olshanski G. and Strahov E. (2006). Giambelli compatible point processes. Adv. in Appl. Math. 37(2): 209–248 zbMATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Borodin A. and Strahov E. (2006). Averages of characteristic polynomials in Random Matrix Theory. Commun. Pure and Applied Math. 59(2): 161–253 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Brezin E. and Hikami S. (2000). Characteristic polynomials of random matrices. Commun. Math. Phys. 214: 111–135 zbMATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Bump D. and Gamburd A. (2006). On the average of characteristic polynomials from classical groups. Commun. Math. Phys. 265: 227–274 zbMATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Conrey J.B., Farmer D.W., Keating J.P., Rubinstein M.O. and Snaith N.C. (2003). Autocorrelation of random matrix polynomials. Commun. Math. Phys. 237: 365–395 zbMATHMathSciNetADSGoogle Scholar
  14. 14.
    Conrey J.B., Forrester P.J. and Snaith N.C. (2005). Averages of ratios of characteristic polynomials for the compact classical groups. Int. Math. Res. Not. 7: 397–431 CrossRefMathSciNetGoogle Scholar
  15. 15.
    Conrey, J.B., Farmer, D.W., Zirnbauer, M.R.: Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups U(N)., 2005
  16. 16.
    Diaconis, P., Gamburd, A.: Random matrices, magic squares and matching polynomials. Electron. J. Combin. 11(2), Research Paper 2, 26 pp. (2004/05)Google Scholar
  17. 17.
    Edelman A. (1997). The probability that a random real gaussian matrix has k real eigenvalues, related distributions and the Cirular law. J. Multiv. Anal. 60: 203–232 zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Edelman A., Kostlan E. and Shub M. (1994). How many eigenvalues of a random matrix are real?. J. Amer. Math. Soc. 7: 247–267 zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Feinberg J. and Zee A. (1997). Non-Gaussian Non-Hermitean Random Matrix Theory: phase transitions and addition formalism. Nucl. Phys. B 501: 643–669 zbMATHCrossRefMathSciNetADSGoogle Scholar
  20. 20.
    Feinberg J., Scalettar R. and Zee A. (2001). “Single Ring Theorem” and the Disk-Annulus Phase Transition. J. Math. Phys. 42: 5718–5740 zbMATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Fyodorov Y.V. (2002). Negative moments of characteristic polynomials of random matrices: Ingham-Siegel integral as an alternative to Hubbard-Stratonovich transformation. Nucl. Phys. B 621: 643–674 zbMATHCrossRefMathSciNetADSGoogle Scholar
  22. 22.
    Fyodorov Y.V. and Akemann G. (2003). On the supersymmetric partition function in QCD-inspired random matrix models. JETP Lett. 77: 438–441 CrossRefADSGoogle Scholar
  23. 23.
    Fyodorov Y.V. and Khoruzhenko B.A. (1999). Systematic analytical approach to correlation functions of resonances in quantum chaotic scattering. Phys. Rev. Let. 83: 65–68 CrossRefADSGoogle Scholar
  24. 24.
    Fyodorov Y.V. and Sommers H.-J. (1997). Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering: Random matrix approach for systems with broken time-reversal invariance. J. Math. Phys. 38: 1918–1981 zbMATHCrossRefMathSciNetADSGoogle Scholar
  25. 25.
    Fyodorov Y.V. and Sommers H.-J. (2003). Random matrices close to Hermitian or unitary: overview of methods and results. J. Phys. A 36: 3303–3347 zbMATHCrossRefMathSciNetADSGoogle Scholar
  26. 26.
    Fyodorov Y.V. and Strahov E. (2003). An exact formula for general spectral correlation function of random Hermitian matrices. J. Phys. A: Maths and General 36: 3203–3213 zbMATHCrossRefMathSciNetADSGoogle Scholar
  27. 27.
    Fyodorov Y.V. and Strahov E. (2002). Characteristic polynomials of random Hermitian matrices and Duistermaat-Heckman localisation on non-compact Kähler Manifolds. Nucl. Phys. B 630: 453–491 zbMATHMathSciNetADSGoogle Scholar
  28. 28.
    Haagerup U. and Larsen F. (2000). Brown’s spectral distribution measure for R-diagonal elements in finite von Neumann algebras. J. Funct. Anal. 176: 331–367 zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Halasz M.A., Jackson A.D. and Verbaarschot J.J.M. (1997). Fermion determinants in matrix models of QCD at nonzero chemical potential. Phys. Rev. D 56: 5140–5152 CrossRefMathSciNetADSGoogle Scholar
  30. 30.
    Hua, L.K.: Harmonic Analysis of Functions of Several Complex variables in the Classical Domains. Providence, RI: Amer. Math. Soc., 1963Google Scholar
  31. 31.
    Ginibre J. (1964). Statistical Ensembles of Complex, Quaternion and Real Matrices. J. Math. Phys. 6: 440–449 CrossRefMathSciNetADSGoogle Scholar
  32. 32.
    Gradshtein, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 5th ed., A. Jeffrey, ed. New York: Academic Press, 1994Google Scholar
  33. 33.
    Kadell K.W.J. (1997). The Selberg-Jack symmetric functions. Adv. Math. 130: 33–102 zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Kaneko J. (1993). Selberg integrals and hypergeometric functions associated with Jack polynomials. SIAM J. Math. Anal. 24: 1086–1110 zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Keating J.P. and Snaith N.C. (2000). Random matrix theory and ζ (1/2 + it). Commun. Math. Phys. 214: 57–89 zbMATHCrossRefMathSciNetADSGoogle Scholar
  36. 36.
    Keating J.P. and Snaith N.C. (2000). Random matrix theory and L-functions at s = 1/2. Commun. Math. Phys. 214: 91–110 zbMATHCrossRefMathSciNetADSGoogle Scholar
  37. 37.
    Macdonald I.G. (1995). Symmetric Functions and Hall Polynomials. 2nd ed. Clarendon Press, Oxford, Oxford zbMATHGoogle Scholar
  38. 38.
    Mehta M.L. (2004). Random Matrices. 3rd ed. Elsevier/Academic Press, Amsterdam zbMATHGoogle Scholar
  39. 39.
    Orlov, A.Yu.: New Solvable Matrix Integrals. In: Proceedings of 6th International Workshop on Conformal Field Theory and Integrable Models. Internat. J. Modern Phys. A 19, May, suppl., 276–293 (2004)Google Scholar
  40. 40.
    Pólya G. and Szegö G. (1972). Problems and Theorems in Analysis. Vol. I, Springer-Verlag, Berlin-Heidelberg-New York zbMATHGoogle Scholar
  41. 41.
    Pólya G. and Szegö G. (1976). Problems and Theorems in Analysis. Vol. II, Springer-Verlag, Berlin-Heidelberg-New York Google Scholar
  42. 42.
    Schlittgen B. and Wettig T. (2003). Generalizations of some integrals over the unitary group. J. Phys. A: Math and General 36: 3195–3202 zbMATHCrossRefMathSciNetADSGoogle Scholar
  43. 43.
    Shuryak E.V. and Verbaarschot J.J.M. (1993). Random matrix theory and spectral sum rules for the Dirac operator in QCD. Nucl. Phys. A 560: 306–320 CrossRefADSGoogle Scholar
  44. 44.
    Strahov, E.: Moments of characteristic polynomials enumerate two-rowed lexicographic arrays. Electron. J. Combin. 10, Research paper 24, 8 pp. (2003)Google Scholar
  45. 45.
    Trotter H.F. (1984). Eigenvalue distributions of large Hermitian matrices: Wigner semicircle and a theorem of Kac, Murdock and Szego. Adv. Math. 54: 67–82 zbMATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Verbaarschot J.J.M. (1994). Spectrum of the QCD Dirac Operator and Chiral Random Matrix Theory. Phys. Rev. Lett. 72: 2531–2533 CrossRefMathSciNetADSGoogle Scholar
  47. 47.
    Verbaarschot, J.J.M.: QCD, chiral random matrix theory and integrability. In: Applications of random matrices in physics, NATO Sci. Ser. II Math. Phys. Chem. 221, Dordrecht: Springer, 2006, pp. 163–217Google Scholar
  48. 48.
    Wilkinson J.H. (1965). The Algebraic Eigenvalue Problem. Clarendon Press, Oxford zbMATHGoogle Scholar
  49. 49.
    Zirnbauer M.R. (1996). Supersymmetry for systems with unitary disorder: circular ensembles. J. Phys. A: Math and General 29: 7113–7136 zbMATHCrossRefMathSciNetADSGoogle Scholar
  50. 50.
    Zyczkowski K. and Sommers H.-J. (2000). Truncations of random unitary matrices. J. Phys. A: Math. and General 33: 2045–2058 zbMATHCrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of NottinghamNottinghamUK
  2. 2.School of Mathematical Sciences, Queen MaryUniversity of LondonLondonUK

Personalised recommendations